Weyl Transforms (Universitext)

Universitext Editorial Board (North America):

S. Axler F.W. Gehring K.A. Ribet

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M.W. Wong

Weyl Transforms

Springer

M.W. Wong Department of Mathematics and Statistics York University Toronto, Ontario M3J 1P3 Canada Editorial Board (North America): S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA

F.W. Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA

K.A. Ribet Department of Mathematics University of California at Berkeley Berkeley, CA 94720-3840 USA

Mathematics Subject Classification (1991): 44A15, 42-01, 43-01

Library of Congress Cataloging-in-Publication Data Wong, M.W. Weyl transforms / M.W. Wong. p. cm. — (Universitext) Includes bibliographical references and indexes. ISBN 0-387-98414-3 (hardcover : alk. paper) l.Pseudodifferential operators. 2. Fourier analysis. I. Title. Qa329.7.W66 1998 515'.7242—dc21 98-13042

© 1998 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

ISBN 0-387-98414-3 Springer-Verlag New York Berlin Heidelberg

SPIN 10663135

Preface

This book is an outgrowth of courses given by me for graduate students at York University in the past ten years. The actual writing of the book in this form was carried out at York University, Peking University, the Academia Sinica in Beijing, the University of California at Irvine, Osaka University, and the University of Delaware. The idea of writing this book was first conceived in the summer of 1989, and the protracted period of gestation was due to my daily duties as a professor at York University. I would like to thank Professor K.C. Chang, of Peking University; Professor Shujie Li, of the Academia Sinica in Beijing; Professor Martin Schechter, of the University of California at Irvine; Professor Michihiro Nagase, of Osaka University; and Professor M.Z. Nashed, of the University of Delaware, for providing me with stimulating environments for the exchange of ideas and the actual writing of the book. We study in this book the properties of pseudo-differential operators arising in quantum mechanics, first envisaged in [33] by Hermann Weyl, as bounded linear operators on L2 (Rn ). Thus, it is natural to call the operators treated in this book Weyl transforms. To be specific, my original plan was to supplement the standard graduate course in pseudo-differential operators at York University by writing a set of lecture notes on the derivation of a formula from first principles for the product of two Weyl transforms. This was achieved in the summer of 1990 when I was visiting Peking University and the Academia Sinica in Beijing. Chapters 2–6 of the book, which appeared then, albeit in embryonic form, already contained the formula for the product of two Weyl transforms obtained by Pool in [20]. Chapters 8 and 9 were written in the summer of 1993 at York University in order to get another formula for the product of two Weyl transforms using relatively new ideas, e.g.,

vi

Preface

the Heisenberg group and the twisted convolution, in noncommutative harmonic analysis developed by Folland in [6] and Stein in [26], among others. The result was an account, given in Chapter 9, of a formula for the product of two Weyl transforms in the paper [10] by Grossmann, Loupias, and Stein. A preliminary version of the derivations of the two formulas was written up for private circulation in the second quarter of 1994–95 at the University of California at Irvine. In the summer of 1994, I gave a course in special topics in pseudo-differential operators tailored to the needs of my Ph.D. students at York University. I chose to study the criteria in terms of the symbols for the boundedness and compactness of the Weyl transforms. Two sets of results were presented. The first set was about the compactness of a Weyl transform with symbol in Lr (R2n ), 1 ≤ r ≤ ∞, and the second set, inspired by the book [29] by Thangavelu, was concerned with the criteria for the boundedness and compactness of Weyl transforms in terms of symbols evaluated at Wigner transforms of Hermite functions. The two sets of results can be found in, respectively, Chapters 11–14 and Chapters 24–27. Chapter 28 is devoted to the study of the eigenvalues and eigenfunctions of a Weyl transform of which the symbol is a Dirac delta on a disk in R2 . The preliminary version of the formulas for the product of two Weyl transforms and the lecture notes of the topics course given in the summer of 1994 were then put together, simplified, polished, and supplemented with background materials at Osaka University and the University of Delaware in the winter of 1997. To this end, I found it instructive to add new chapters, i.e., Chapters 15–17, on localization operators initiated by Daubechies in [3, 4] and Daubechies and Paul in [5], and the closely related theory of square-integrable group representations studied by Grossmann, Morlet, and Paul in [11, 12]. The final two chapters were added in an attempt to make explicit the role of the symplectic group in the study of Weyl transforms. The connections of the Weyl transforms with quantization in physics, highlighted in this book, can be found in the references [6, 10, 20, 26, 33] already cited, the book [2] by Berezin and Shubin, the paper [18] by Iancu and Wong, and the papers [37, 38] by Wong. All the topics in this book should be accessible to a first-year graduate student. The book is a natural sequel to a first course in pseudo-differential operators, but no familiarity with even the basics of pseudo-differential operators is required for a good understanding of the entire book. The only essential prerequisites are the elementary properties of the Fourier transform and tempered distributions given in the beginning chapters of, say, the book [8] by Goldberg, the book [27] by Stein and Weiss, and the book [36] by Wong, and these are collected in Chapter 1. Of course, a nodding acquaintance with basic functional analysis is necessary for an intelligent reading of this book. Finally, it must be emphasized that this book is far from being a definitive treatise on Weyl transforms. Thus, the choice of topics in this book was guided by personal predilections, and the references at the end of the book are limited to those that have been instrumental in my understanding of Weyl transforms.

Contents

Preface

v

1

Prerequisite Topics in Fourier Analysis

1

2

The Fourier–Wigner Transform

9

3

The Wigner Transform

13

4

The Weyl Transform

19

5

Hilbert–Schmidt Operators on L2 (Rn )

25

6

The Tensor Product in L2 (Rn )

29

7 H ∗ -Algebras and the Weyl Calculus

33

8

The Heisenberg Group

37

9

The Twisted Convolution

43

10 The Riesz–Thorin Theorem

47

11 Weyl Transforms with Symbols in Lr (R2n ), 1 ≤ r ≤ 2

55

12 Weyl Transforms with Symbols in L∞ (R2n )

59

viii

Contents

13 Weyl Transforms with Symbols in Lr (R2n ), 2 < r < ∞

63

14 Compact Weyl Transforms

71

15 Localization Operators

75

16 A Fourier Transform

79

17 Compact Localization Operators

83

18 Hermite Polynomials

87

19 Hermite Functions

93

20 Laguerre Polynomials

95

21 Hermite Functions on C

101

22 Vector Fields on C

103

23 Laguerre Formulas for Hermite Functions on C

107

24 Weyl Transforms on L2 (R) with Radial Symbols

113

25 Another Fourier Transform

119

26 A Class of Compact Weyl Transforms on L2 (R)

123

27 A Class of Bounded Weyl Transforms on L2 (R)

127

28 A Weyl Transform with Symbol in S  (R2 )

131

29 The Symplectic Group

135

30 Symplectic Invariance of Weyl Transforms

145

Notation Index

155

Index

157

1 Prerequisite Topics in Fourier Analysis

The basic topics in Fourier analysis that we need for a good understanding of the book are collected in this chapter. In view of the fact that these topics can be found in many books on Fourier analysis, e.g., [8] by Goldberg, [27] by Stein and Weiss, and [36] by Wong, among others, we provide only the proofs of the key results in the study of the Weyl transform. Another important role played by this chapter is to fix the notation used throughout the book. Let Rn  {(x1 , x2 , . . . , xn ) : xj real numbers}. Points in Rn are denoted by x, y, ξ, η, etc. Let x  (x1 , x2 , . . . , xn ) and y  (y1 , y2 , . . . , yn ) be in Rn . The inner product x · y of x and y is defined by x·y 

n


xj yj ,

j 1

and the norm |x| of x is defined by  |x| 

n


 21 xj2

.

j 1

We denote the differential operators ∂x∂ 1 , ∂x∂ 2 , . . . , ∂x∂ n on Rn by ∂1 , ∂2 , . . . , ∂n , respectively, and the differential operators −i∂1 , −i∂2 , . . . , −i∂n on Rn by D1 , D2 , . . . , Dn , respectively, where i 2  −1. A reason for using the factor of −i is to make some formulas, e.g., Proposition 1.8, look better, but the main justification for its appearance lies in the fact that the quantum-mechanical momentum observable in the direction of the j th coordinate is represented by Dj if we choose to work with units that give the value 1 to Planck’s constant. More de-

2

1. Prerequisite Topics in Fourier Analysis

tails are given in the discussion at the end of Chapter 4. A linear partial differential operator P (x, D) on Rn is given by
P (x, D)  aα (x)D α , x ∈ Rn , |α|≤m

where α  (α1 , α2 , . . . , αn ) is a multi-index, i.e., an n-tuple of nonnegative inn α tegers; |α|   D α1 D α2 · · · D αn , and aα is j 1 αj is the length of α; D n a measurable complex-valued function on R for |α| ≤ m. The symbol of the differential operator P (x, D) is the function on R2n defined by
P (x, ξ )  aα (x)ξ α , x, ξ ∈ Rn , |α|≤m

ξ1α1 ξ2α2

where ξ α  · · · ξnαn . The differential operator ∂ α , for any multi-index α, defined by ∂ α  ∂1α1 ∂2α2 · · · ∂nαn , will also be used frequently in the book. We write ∂xα (or ∂ξα ) for ∂ α , and Dxα (or Dξα ) for D α , when we need to specify the variable with respect to which we differentiate. Let f and g be infinitely differentiable functions on Rn . Then we have the Leibnitz formula
α  α D (f g)  (D β f )(D α−β g) β β≤α for all multi-indices α, and the more general Leibnitz formula
P (D)(f g)  (P (µ) (D)f )(D µ g) |µ|≤m

 for any linear partial differential operator P (D)  |α|≤m aα D α with constant        coefficients, where β ≤ α means βj ≤ αj , j  1, 2, · · · , n, βα  αβ11 αβ22 . . . αβnn , µ!  µ1 !µ2 ! · · · µn !, and P (µ) (D) is the linear partial differential operator with symbol P (µ) on Rn given by P (µ) (ξ )  (∂ µ P )(ξ ),

ξ ∈ Rn .

Now we let C0∞ (Rn ) be the set of all infinitely differentiable functions on Rn with compact supports, and we let S(Rn ) be the set of all infinitely differentiable functions on Rn such that sup |x α (∂ β ϕ)(x)| < ∞

x∈Rn

for all multi-indices α and β. Theorem 1.1. C0∞ (Rn ) and S(Rn ) are dense in Lr (Rn ), 1 ≤ r < ∞. Theorem 1.1 can be proved using the convolution of functions on Rn .

1. Prerequisite Topics in Fourier Analysis

3

Theorem 1.2. (Young’s Inequality) Let f ∈ L1 (Rn ) and g ∈ Lr (Rn ), 1 ≤ r ≤ ∞. Then the integral f (x − y)g(y)dy Rn

exists for almost all x in R . If we denote the value of the integral by (f ∗ g)(x), then f ∗ g ∈ Lr (Rn ) and n

f ∗ gLr (Rn ) ≤ f L1 (Rn ) gLr (Rn ) . Remark 1.3. The function f ∗ g in Theorem 1.2 is usually called the convolution of f and g. The formulation of Young’s inequality for the convolution of sequences ∞ {ak }∞ k−∞ and {bk }k−∞ of complex numbers is left as an exercise. It is useful to have the following result. Proposition 1.4. Let f and g be in S(Rn ). Then f ∗ g ∈ S(Rn ). Our next result is a technique of regularization to be used in the proofs of Theorems 1.11, 3.1, and 16.1.

Theorem 1.5. Let ϕ ∈ L1 (Rn ) be such that Rn ϕ(x)dx  a. For any positive number ε, we define the function ϕε on Rn by x , x ∈ Rn . ϕε (x)  ε−n ϕ ε Then, for any bounded function f on Rn that is continuous on an open subset V of Rn , f ∗ ϕε → af uniformly on compact subsets of V as ε → 0. Proof. Without loss of generality, we can assume that ϕ(x)  0 for almost all x in Rn . Since ϕε (x)dx  a Rn

for any positive number ε, it follows that {f (x − εy) − f (x)}ϕ(y)dy (f ∗ ϕε )(x) − af (x) 

(1.1)

Rn

for all x in Rn . Let K be a compact subset of V and δ be a positive number. Let K1 be a compact subset of V such that K is properly contained in K1 . Then there exists a positive number δ1 such that δ ϕ−1 (1.2) L1 (Rn ) 2 for all x and y in K1 with |x − y| < δ1 . Let U be an open subset of Rn such that K ⊂ U ⊂ K1 . Let δ2 be the distance between K and the complement of U in Rn and let δ0  min(δ1 , δ2 ). Then

K⊂ B(x, δ0 ) ⊂ U, (1.3) |f (x) − f (y)|
1 implies that W (f )(x, ξ )  0 for all ξ in Rn . Now, by (13.19),  |W (f )(x, ξ )|r dx dξ < ∞. (13.21) n

Rn

Rn

Then, for any real-valued function θ on R2n , we get, by (13.20) and (13.21),   r   r1   iθ (x,ξ )  W (f )(x, ξ )e dx  dξ  n R



≤

|x|≤1



|x|≤1

Rn

 ≤ Thus, by (13.22),

r

|W (f )(x, ξ )| dξ

 1r 

|x|≤1

dx

|x|≤1

Rn



 1 r

dx r

Rn

|W (f )(x, ξ )| dx dξ



W (f )(x, ξ )eiθ (x,ξ ) dx ∈ Lr (Rn )

 1 r

< ∞.

(13.22)

(13.23)

68

13. Weyl Transforms with Symbols in Lr (R2n ), 2 < r < ∞

as a function of ξ on Rn . Now, let θ(x, ξ )  2x · ξ, x, ξ ∈ Rn .

(13.24)

Then, by (3.12) and (13.24), W (f )(x, ξ )eiθ (x,ξ ) dx |x|≤1

  p  p e  (2π) dp eiθ (x,ξ ) dx f x+ f x− 2 2 Rn Rn  p p  p −n/2  (2π) e2iξ ·(x− 2 ) f x + f x− dx dp (13.25) 2 2 Rn Rn −n/2





−iξ ·p

for all ξ in Rn . Let u  x + p2 and v  x − p2 in (13.25). Then, using the Fourier inversion formula, W (f )(x, ξ )eiθ (x,ξ ) dx Rn  (2π )−n/2 e2iξ ·v f (u)f (v)du dv  (2π)−n/2



Rn

Rn

Rn



f (u)

Rn

 (2π )n/2 fˆ(0)fˆ(2ξ ),

 e−2iξ ·v f (v)dv du

ξ ∈ Rn .

(13.26)

Now, let Q  {x ∈ Rn : −a ≤ xj ≤ a, j  1, 2, . . . , n} be a cube lying inside {x ∈ Rn : |x| ≤ 1}. Let α ∈ (0, 21 ) and let f be the function on Rn defined by  n −α j 1 |xj | , x ∈ Q, xj  0, j  1, 2, . . . , n, f (x)  0, otherwise. Then fˆ(ξ )  (2π)−n/2



e−ix·ξ Q

 (2π)−n/2 Now, for ξj > 0, a −a

n 

n 

|xj |−α dx

j 1 a

j 1 −a

e−ixj ξj |xj |−α dxj 

e−ixj ξj |xj |−α dxj ,



a

−a



ξ ∈ Rn .

cos(xj ξj )|xj |−α dxj a

cos(xj ξj )xj−α dxj 0  aξj  −α 2 t cos t dt ξj−1+α . 2

(13.27)

0

(13.28)

13. Weyl Transforms with Symbols in Lr (R2n ), 2 < r < ∞

69

By Lemma 13.4, there is a positive constant A such that  aξj    −α  t cos t dt  ≥ A 

(13.29)

for ξj large enough. Thus, by (13.27), (13.28), and (13.29),   n  −1+α −n/2 |fˆ(ξ )| ≥ (2π) 2 n An ξj

(13.30)

0

j 1

whenever ξ1 , ξ2 , . . . , ξn are all larger than some positive number R. Hence, by (13.30), n ∞   nr     |fˆ(ξ )|r dξ ≥ (2π)− 2 2nr Anr ξj(−1+α)r dξj  ∞ (13.31) Rn

j 1 R

if (1 − α)r  < 1. Therefore, n

Wf Lr  (R2n ) ≤ (2π ) 2 Cf 2L2 (Rn )

(13.32)

is impossible if (1 − α)r  < 1, and (13.33) is obtained if we pick α to be some number in

(13.33) ( 1r , 21 ).

2

14 Compact Weyl Transforms

In view of the positive result, i.e., Theorem 11.1, and the negative results, i.e., Theorem 12.4 and Simon’s Theorem 13.1, it is of genuine interest to have a sufficient condition on a function σ in Lr (R2n ), 2 ≤ r ≤ ∞, such that the Weyl transform Wσ , defined by (12.1), is a compact operator from L2 (Rn ) into L2 (Rn ). To do this, we introduce the space Lr∗ (R2n ), 1 ≤ r ≤ ∞, defined by 

Lr∗ (R2n )  {σ ∈ Lr (R2n ) : σˆ ∈ Lr (R2n )}.

(14.1)

To gain some insight into the space Lr∗ (R2n ), 1 ≤ r ≤ ∞, we use the following theorem. Theorem 14.1. (The Hausdorff–Young Inequality) The Fourier transform is a  bounded linear operator from Lr (Rn ) into Lr (Rn ), 1 ≤ r ≤ 2. In fact, n 2 fˆLr  (Rn ) ≤ (2π)− 2 (1− r  ) f Lr (Rn ) ,

Proof.

f ∈ Lr (Rn ).

(14.2)

By the Plancherel theorem, we get fˆL2 (Rn )  f L2 (Rn ) ,

f ∈ L2 (Rn ),

(14.3)

and using the definition of the Fourier transform, we get fˆL∞ (Rn ) ≤ (2π )−n/2 f L1 (Rn ) ,

f ∈ L1 (Rn ).

(14.4)

Let β1  21 , α1  21 , β2  0, and α2  1. Let M1  1 and M2  (2π )−n/2 . Let θ  1 − r2 . Then, by (14.3), (14.4), and the Riesz–Thorin theorem, fˆLr  (Rn ) ≤ (2π )− 2 (1− r  ) f Lr (Rn ) , n

2

f ∈ D,

(14.5)

72

14. Compact Weyl Transforms

where D is the set of all simple functions f on Rn such that the Lebesgue measure of the set {x ∈ Rn : f (x)  0} is finite. Since D is dense in Lr (Rn ), a limiting argument can be used to complete the proof that the Fourier transform is a bounded linear operator from Lr (Rn ) into  Lr (Rn ) and (14.2) is valid. 2 As an immediate consequence of the Hausdorff–Young inequality, we give the following corollary. Corollary 14.2. Lr∗ (R2n )  Lr (R2n ), 1 ≤ r ≤ 2. We can now give a result supplementing Theorems 11.1 and 11.3. Theorem 14.3. Let σ ∈ Lr∗ (R2n ), 2 ≤ r ≤ ∞. Then Wσ : L2 (Rn ) → L2 (Rn ) is a bounded linear operator. In fact, Wσ ∗ ≤ (2π)− r  σˆ Lr  (R2n ) , n

σ ∈ Lr∗ (R2n ).

(14.6)

Furthermore, Wσ : L2 (Rn ) → L2 (Rn ) is a compact operator. To prove Theorem 14.3, we need some lemmas. Lemma 14.4. Let f and g be in S(Rn ). Then ˜ 2x), W (f, g)(x, ξ )  2n V (f, g)(−2ξ, Proof.

Let q 

x, ξ ∈ Rn .

p 2

in (3.12). Then, by Proposition 2.3,  p  p e−iξ ·p f x + g x− dp W (f, g)(x, ξ )  (2π )−n/2 2 2 Rn e−2iξ ·q f (x + q)g(x − q)dq  2n (2π)−n/2 Rn  2n (2π)−n/2 e−2iξ ·q f (q + x)g(q ˜ − x)dq Rn

 2n V (f, g)(−2ξ, ˜ 2x),

x, ξ ∈ Rn , 2

and the proof is complete. Lemma 14.5. Let σ ∈ by

Lr∗ (R2n ),

1 ≤ r ≤ ∞. Then the function τ on R

2n

defined

τ (x, ξ )  σ (ξ, x),

x, ξ ∈ Rn ,

(14.7)

τˆ (q, p)  σˆ (p, q),

q, p ∈ Rn .

(14.8)

is also in Lr∗ (R2n ), and To prove Lemma 14.5, we need another lemma. Lemma 14.6. Let f and g be in Lr (Rn ), 1 ≤ r ≤ 2. Then f (x)g(x)dx ˆ  fˆ(x)g(x)dx. Rn

Rn

14. Compact Weyl Transforms

73

∞ Proof. Let f and g be in Lr (Rn ), 1 ≤ r ≤ 2. Then we let {fk }∞ k1 and {gk }k1 be sequences of functions in S(Rn ) such that fk → f and gk → g in Lr (Rn ) as k → ∞. Then, by the Hausdorff–Young inequality, fˆk → fˆ and gˆ k → gˆ in  Lr (Rn ) as k → ∞. Hence, by Proposition 1.10, f (x)g(x)dx ˆ  lim fk (x)gˆ k (x)dx k→∞ Rn Rn 2 fˆk (x)gk (x)dx  fˆ(x)g(x)dx.  lim k→∞ Rn

Rn

Proof of Lemma 14.5. For any ϕ in S(R2n ), we get, by Lemma 14.6, τˆ (ϕ)  τ (ϕ) ˆ τ (x, ξ )ϕ(x, ˆ ξ )dx dξ  n n R R ˆ τ (x, ξ )ψ(ξ, x)dx dξ,  Rn

(14.9)

Rn

where ψ(q, p)  ϕ(p, q),

q, p ∈ Rn .

Thus, by (14.7), (14.9), (14.10), and Lemma 14.6, ˆ σ (ξ, x)ψ(ξ, x)dξ dx τˆ (ϕ)  Rn Rn σˆ (p, q)ψ(p, q)dq dp  n n R R σˆ (p, q)ϕ(q, p)dq dp,  Rn

Rn

ϕ ∈ S(R2n ).

(14.10)

(14.11)



Thus, by (14.11), τˆ ∈ Lr (R2n ), and (14.8) is valid.

2

Proof of Theorem 14.3. Let f and g be in S(Rn ). Then, by (12.1), Theorem 3.1, (3.12), and Lemmas 14.4–14.6, ¯  (2π )−n/2 σ (W (f, g)) (Wσ f )(g) σ (x, ξ )W (f, g)(x, ξ )dx dξ  (2π )−n/2 n n R R σ (x, ξ )V (f, g)ˆ(x, ξ )dx dξ  (2π)−n/2 Rn Rn σˆ (q, p)V (f, g)(q, p)dq dp  (2π )−n/2 Rn Rn p q ,− σˆ (q, p)W (f, g) ˜ dq dp  2−n (2π )−n/2 n n 2 2 R R σˆ (−2q, 2p)W (f, g)(p, ˜ q)dq dp  2n (2π)−n/2 Rn

Rn

74

14. Compact Weyl Transforms

 2n (2π)−n/2  2n (2π)−n/2  2n (2π)−n/2





Rn

σˆ (−2p, 2q)W (f, g)(q, ˜ p)dq dp

Rn





R R n

n

Rn

Rn

τˆ (2q, −2p)W (f, g)(q, ˜ p)dq dp (δ τˆ )(q, p)W (f, g)(q, ˜ p)dq dp,

(14.12)

where (δ τˆ )(q, p)  τˆ (2q, −2p), r

q, p ∈ Rn .

(14.13)

Since δ τˆ ∈ L (R ), it follows from Theorem 11.1, (14.12), and (14.13) that 2n

(Wσ f )(g) ¯  2n Wδτˆ f, g , ˜ and hence |(Wσ f )(g)| ¯ ≤

f, g ∈ S(Rn ),

  n 2 r δ τˆ L2 (R2n ) f L2 (Rn ) gL2 (Rn ) π

(14.14)

for all f and g in S(Rn ). Therefore, by (14.14), Wσ : L2 (Rn ) → L2 (Rn ) is a bounded linear operator and   n 2 r Wσ ∗ ≤ δ τˆ Lr  (R2n ) . (14.15) π But, by (14.13),  δ τˆ Lr  (R2n ) 



Rn

  2

Rn − 2n r





Rn

|τˆ (2q, −2p)|r dq dp 

Rn

 1

|σˆ (−2p, 2q)|r dq dp

r

 1

σˆ Lr  (R2n ) ,

r

(14.16)

and hence (14.6) follows from (14.15) and (14.16). To prove that Wσ : L2 (Rn ) → 2n L2 (Rn ) is compact, let {τk }∞ k1 be a sequence of functions in S(R ) such that r 2n 2n τk → σˆ in L (R ) as k → ∞. For k  1, 2, . . . , let σk ∈ S(R ) be such that σˆ k  τk . Then, by (14.6), Wσ − Wσk ∗ ≤ (2π )− r  σˆ − σˆ k Lr  (R2n ) n

 (2π)− r  σˆ − τk Lr  (R2n ) , n

and hence Wσk → Wσ in B(L2 (Rn )) as k → ∞. By Theorem 6.8, Wσk is a Hilbert–Schmidt operator on L2 (Rn ), and hence, by Lemma 11.4, compact, for k  1, 2, . . . . Thus, Wσ : L2 (Rn ) → L2 (Rn ) is compact. 2

15 Localization Operators

Our aim for Chapters 15–17 is to show that the localization operators introduced by Daubechies in the paper [3] as filters in signal analysis are examples of Weyl transforms that enjoy good mapping properties as compact operators from L2 (Rn ) into L2 (Rn ). In this chapter, we define the notion of a localization operator with symbol in Lr (R2n ), 1 ≤ r ≤ ∞, and prove that a localization operator with symbol in Lr (R2n ), 1 ≤ r ≤ ∞, is a bounded linear operator from L2 (Rn ) into L2 (Rn ). Let Z be the set of all integers. Then Cn × R/2πZ is a locally compact and Hausdorff group in which the group law is given by (q1 , p1 , t1 ) · (q2 , p2 , t2 )  (q1 + q2 , p1 + p2 , t1 + t2 + q1 · p2 ) for all (q1 , p1 , t1 ) and (q2 , p2 , t2 ) in Cn × R/2π Z, where q1 · p2 is the Euclidean inner product of q1 and p2 in Rn ; t1 +t2 and t1 +t2 +q1 ·p2 are cosets in the quotient group R/2πZ in which the group law is addition modulo 2π . On Cn × R/2π Z, the left Haar measure coincides with the right Haar measure and can be identified with the Lebesgue measure dq dp dt on the measurable space Cn × R/2π Z. The locally compact Hausdorff space Cn × R/2π Z endowed with the left (and right) Haar measure dq dp dt is hence unimodular. It is called the Weyl–Heisenberg group, and we denote it by (W H )n . Let π : (W H )n → B(L2 (Rn )) be the mapping defined by 1

(π(q, p, t)f )(x)  ei(p·x+ 2 q·p+t) f (x − q),

x ∈ Rn ,

for all (q, p, t) in (W H )n and f in L2 (Rn ). That π : (W H )n → B(L2 (Rn )) is an irreducible unitary representation is left as an exercise.

76

15. Localization Operators

Let ϕ be the function on Rn defined by ϕ(x)  π − 4 e− n

|x|2 2

,

x ∈ Rn .

(15.1)

Then ϕL2 (Rn )  1,

(15.2)

and it is also an easy exercise to prove that the number cϕ defined by 2π | ϕ, π(q, p, t)ϕ |2 dq dp dt cϕ  Rn

0

Rn

is finite, and in fact cϕ  (2π)n+1 .

(15.3)

The function ϕ is called an admissible wavelet for the irreducible unitary representation π : (W H )n → B(L2 (Rn )), and the representation π : (W H )n → B(L2 (Rn )) is called square integrable. It is left as an exercise that we have the resolution of the identity formula, i.e., 1 2π f, π(z, t)ϕ π(z, t)ϕ, g dz dt (15.4) f, g  cϕ 0 Cn for all f and g in L2 (Rn ). The theory of the representation π : (W H )n → B(L2 (Rn )) hitherto described is an important, albeit special, case of the theory of square-integrable representations studied in the papers [11,12] by Grossmann, Morlet, and Paul, and the book [17] by Holschneider, among others. For q and p in Rn , we define the function ϕq,p on Rn by ϕq,p (x)  eip·x ϕ(x − q),

x ∈ Rn .

(15.5)

Let F be a measurable function on Cn . Then we define σ : (W H )n → C by σ (z, t)  F (z),

z ∈ Cn , t ∈ [0, 2π ].

(15.6)

Let F be in L1 (Cn ) or L∞ (Cn ). Then, for any f in L2 (Rn ), we define the function LF f on Rn by 1 2π LF f, g  σ (z, t) f, π(z, t)ϕ π(z, t)ϕ, g dz dt cϕ 0 Cn or, in view of (15.3) and (15.6), F (z) f, ϕz ϕz , g dz, LF f, g  (2π )−n Cn

g ∈ L2 (Rn ).

(15.7)

We have the following result. Proposition 15.1. Let F ∈ L1 (Cn ). Then LF : L2 (Rn ) → L2 (Rn ) is a bounded linear operator, and LF ∗ ≤ (2π)−n F L1 (Cn ) .

(15.8)

15. Localization Operators

Proof.

77

Since by (15.2) and (15.5), ϕz L2 (Rn )  1,

z ∈ Cn ,

(15.9)

it follows from (15.9) that for all z in Cn , | f, ϕz ϕz , g | ≤ f L2 (Rn ) gL2 (Rn ) ,

f, g ∈ L2 (Rn ).

(15.10)

Since F ∈ L1 (Cn ), it follows from (15.7) and (15.10) that | LF f, g | ≤ (2π )−n F L1 (Cn ) f L2 (Rn ) gL2 (Rn ) for all f and g in L2 (Rn ) and hence LF : L2 (Rn ) → L2 (Rn ) is a bounded linear operator and (15.8) is valid. 2 We also have the following. Proposition 15.2. Let F ∈ L∞ (Cn ). Then LF : L2 (Rn ) → L2 (Rn ) is a bounded linear operator, and LF ∗ ≤ F L∞ (Cn ) .

(15.11)

Proof.

Let f and g be in L2 (Rn ). Then, by (15.7),  21   −n 2 | f, ϕz | dz | LF f, g | ≤ (2π) F L∞ (Cn ) Cn

| ϕz , g | dz 2

Cn

 21 . (15.12)

But using the resolution of the identity formula, i.e., (15.4), 2 −n f L2 (Rn )  (2π) | f, ϕz |2 dz

(15.13)

Cn

and g2L2 (Rn )

 (2π)

−n

Cn

| g, ϕz |2 dz.

(15.14)

So, by (15.12), (15.13), and (15.14), | LF f, g | ≤ F L∞ (Cn ) f L2 (Rn ) gL2 (Rn ) , 2

and the proof is complete.

Remark 15.3. The bounded linear operators LF : L (R ) → L (R ) already introduced, and to be introduced, in this chapter are called localization operators. This terminology appears to be first used in the papers [3, 4] by Daubechies and [5] by Daubechies and Paul. To understand the terminology better, let us note that when F is identically equal to 1 on Cn , the resolution of the identity formula, i.e., (15.4), implies that the corresponding operator LF is equal to the identity. Thus, the role of the “symbol” F is to assign different weights to different parts of the phase space Cn , i.e., localize on Cn , in order to produce a mathematically interesting operator with applications in various disciplines in science and engineering. To wit, applications to signal analysis can be found in the above-mentioned papers, the paper [14] by He, and the paper [16] by He and Wong. 2

n

2

n

78

15. Localization Operators

We can now associate a localization operator LF : L2 (Rn ) → L2 (Rn ) to every F in Lr (Cn ), 1 < r < ∞. The main result is the following theorem. Theorem 15.4. Let F ∈ Lr (Cn ), 1 < r < ∞. Then there exists a unique bounded linear operator LF : L2 (Rn ) → L2 (Rn ) such that LF ∗ ≤ (2π)− r F Lr (Cn ) , n

(15.15)

and LF f, g , for all f and g in L2 (Rn ), is given by (15.7) for all simple functions F on Cn for which the Lebesgue measure of the set {z ∈ Cn : F (z)  0} is finite. Proof. Let D be the set of all simple functions F on Cn such that the Lebesgue measure of the set {z ∈ Cn : F (z)  0} is finite. Let f ∈ L2 (Rn ) and let T be the linear transformation from D into the set of all Lebesgue-measurable functions on Rn defined by T F  LF f,

F ∈ D.

(15.16)

Then, by (15.8), (15.11), (15.16), and the Riesz–Thorin theorem, LF f L2 (Rn ) ≤ (2π )− r F Lr (Cn ) f L2 (Rn ) n

(15.17)

for all F in D. Hence (15.15) follows immediately from (15.17). Now, let F ∈ Lr (Cn ). Then there exists a sequence {Fk }∞ k1 of functions in D such that Fk → F in Lr (Cn ) as k → ∞. Thus, by (15.15), LFk − LFj ∗ ≤ (2π )− r Fk − Fj Lr (Cn ) → 0 n

∞. Therefore, by (15.18), {LFk }∞ k1

(15.18)

as k, j → is a Cauchy sequence in B(L2 (Rn )). Thus, there is a bounded linear operator LF : L2 (Rn ) → L2 (Rn ) such that LFk → LF in B(L2 (Rn )) as k → ∞. That the limit is independent of the choice of the 2 n 2 n sequence {Fk }∞ k1 in D and that LF : L (R ) → L (R ) so defined is the unique bounded linear operator satisfying the conclusions of the theorem should, by now, be obvious, or if not, it follows from a standard argument, which we leave as an exercise. 2

16 A Fourier Transform

In order to study localization operators in some detail, we first compute the Fourier transform of a function on Cn . Theorem 16.1. Let ϕ be the function on Rn defined by (15.1), and for any x in Rn and f in S(Rn ), let fx be the function on Cn defined by fx (z)  f, ϕz ϕz (x),

z ∈ Cn .

(16.1)

Then fˆx (ζ )  e−

|ζ |2 4

(ρ(−ζ )f )(x),

ζ ∈ Cn ,

(16.2)

where ρ(ζ ), for any ζ in Cn , is given by (2.1). Proof.

Let ε be any positive number. Then we define the function Iε on Cn by ε2 |p|2 Iε (ζ )  (2π )−n e−i(q·ξ +p·η) e− 2 f, ϕz ϕz (x)dz (16.3) Rn

Rn

for all ζ  ξ + iη in Cn , where z  q + ip is a point in Cn . Then, by (15.1), (16.3), and the fact that the Fourier transform of the function ψ on Rn given by ψ(x)  e−

|x|2 2

,

x ∈ Rn ,

(16.4)

is equal to itself, we get (2π)n Iε (ζ )   2 2 −i(q·ξ +p·η) − ε |p| −ip·y 2  e e f (y)e ϕ(y − q)dy eip·x ϕ(x − q)dq dp Rn Rn

Rn

80

16. A Fourier Transform





−iq·ξ

−ip·(η+y−x) − ε

2 |p|2

 dp dq

f (y)ϕ(y − q)ϕ(x − q) e e Rn |η+y−x|2  (2π)n/2 e−iq·ξ f (y)ϕ(y − q)ϕ(x − q)ε −n e− 2ε2 dq n n   R R |η+y−x|2  (2π)n/2 e−iq·ξ ϕ(x − q) ε −n e− 2ε2 f (y)ϕ(y − q)dy dq Rn Rn

e

Rn

2

(16.5)

Rn

for all ζ  ξ + iη in Cn . Now, for each q in Rn , we define Fq : Rn → C by Fq (y)  f (y)ϕ(y − q),

y ∈ Rn .

Then, by (16.4), (16.5), and (16.6), Iε (ζ )  (2π)−n/2 e−iq·ξ ϕ(x − q)(Fq ∗ ψε )(x − η)dq

(16.6)

(16.7)

Rn

for all ζ  ξ + iη in Cn , where ψε (x)  ε−n ψ

x

(16.8) , x ∈ Rn , ε and ψ is the function on Rn given by (16.4). Now, for each fixed q in Rn , we get, by (16.4), (16.6), and (16.8),   n Fq ∗ ψε → (16.9) ψ(x)dx Fq  (2π ) 2 Fq Rn

uniformly on compact subsets of Rn as ε → 0. Furthermore, there exists a positive constant C such that |(Fq ∗ ψε )(w)| ≤ Fq L∞ (Rn ) ψε L1 (Rn ) ≤ sup (|f (y)ϕ(y − q)|)ψL1 (Rn ) y∈Rn

≤ C, w, q ∈ Rn .

(16.10)

So, by (16.6), (16.7), (16.9), (16.10), and the Lebesgue dominated convergence theorem,   lim Iε (ζ )  e−iq·ξ ϕ(x − q)ϕ(x − η − q)dq f (x − η) (16.11) ε→0

Rn

for all ζ  ξ + iη in Cn . But, using (16.1), (16.3), and the Lebesgue dominated convergence theorem, lim Iε (ζ )  (2π )−n e−i(q·ξ +p·η) f, ϕz ϕz (x)dz  fˆx (ζ ) (16.12) ε→0

Rn

Rn

for all ζ  ξ + iη in C . Thus, by (16.11), (16.12), (2.1), and Proposition 2.3,   −iq·ξ ˆ fx (ζ )  e ϕ(x − q)ϕ(x − η − q)dq f (x − η) Rn   −i(x−q)·ξ  e ϕ(q)ϕ(q − η)dq f (x − η) n

Rn

16. A Fourier Transform

 e−ix·ξ f (x − η)



81

 η η η  ϕ q− dq ei(q+ 2 )·ξ ϕ q + 2 2 Rn

 e−ix·ξ e 2 iξ ·η f (x − η)(2π) 2 V (ϕ, ϕ)(ξ, η) n

1

n

 (ρ(−ξ, −η)f )(x)(2π) 2 V (ϕ, ϕ)(ξ, η).

(16.13)

But by Proposition 2.3, (15.1), and the fact that the Fourier transform of the function ψ on Rn defined by (16.4) is equal to itself, we get  η  η e−iy·ξ ϕ y + V (ϕ, ϕ)(ξ, η)  (2π)−n/2 ϕ y− dy 2 2 Rn η 2 η 2 n 1 1  (2π)−n/2 π − 2 eiy·ξ e− 2 |y+ 2 | e− 2 |y− 2 | dy n R 1 2 |η|2 −n/2 − n2  (2π ) π eiy·ξ e− 2 {2|y| + 2 } dy Rn |η|2 n 2 eiy·ξ e−|y| dy  π − 2 e− 4 (2π)−n/2 Rn 2 |y|2 i √y ·ξ − n2 − |η|4 − n2 −n/2 π e 2 (2π) e 2 e− 2 dy Rn

 (2π)−n/2 e

2 − |η|4

e

2 − |ξ4|

 (2π)−n/2 e−

|ζ |2 4

for all ζ  ξ + iη in C . Thus, (16.2) follows from (16.13) and (16.14). n

(16.14) 2

17 Compact Localization Operators

The aim of this chapter is to use the Fourier transform computed in Chapter 16 to prove that every localization operator is a Weyl transform and the fact that a localization operator with symbol in Lr (R2n ), 1 ≤ r < ∞, is compact. Theorem 17.1. Let  be the function on Cn defined by (z)  π −n e−|z| , 2

z ∈ Cn .

(17.1)

Then, for all F in Lr (Cn ), 1 ≤ r ≤ ∞, the Weyl transform WF ∗ , initially defined on S(Rn ), can be extended to a bounded linear operator from L2 (Rn ) into L2 (Rn ) that is equal to LF . Proof. We begin with the case when F is in S(Cn ). Then, for all f in S(Rn ), we get, by (15.7) and (16.1), −n F (z)fx (z)dz (LF f )(x)  (2π ) n C  (2π)−n (17.2) Fˇ (ζ )fˆx (ζ ) dζ, x ∈ Rn , Cn

where Fˇ is the inverse Fourier transform of F . Thus, by (17.2), 1 2 (LF f )(x)  (2π )−n Fˇ (ζ )e− 4 |ζ | (ρ(−ζ )f )(x)dζ Cn 1 2  (2π)−n Fˆ (ζ )e− 4 |ζ | (ρ(ζ )f )(x)dζ, x ∈ Rn . Cn

(17.3)

84

17. Compact Localization Operators

But, by (9.4), (Wσ f )(x)  (2π)−n

Cn

σˆ (ζ )(ρ(ζ )f )(x)dζ, x ∈ Rn ,

(17.4)

for any σ in S(Cn ). Then, by (17.1), (17.3), (17.4), and Proposition 1.7, LF  WF ∗ .

(17.5)

Now, let F ∈ Lr (Cn ), 1 ≤ r < ∞. Then there exists a sequence {Fk }∞ k1 of functions in S(Cn ) such that Fk → F in Lr (Cn ) as k → ∞. Thus, by Proposition 15.1, Theorem 15.4, and (17.5), WFk ∗  LFk → LF

(17.6)

in B(L (R )) as k → ∞. But for all f and g in S(R ), we get, by (12.1) and (17.6), 2

n

n

(WFk ∗ f )(g)  (2π)−n/2 (Fk ∗ )(W (f, g)) ¯ → (2π )−n/2 (F ∗ )(W (f, g)) ¯  (WF ∗ f )(g)

(17.7)

as k → ∞. Thus, for all f in S(Rn ), we get, by (17.7), WFk ∗ f → WF ∗ f

(17.8)

in S  (Rn ) as k → ∞. But of course, by (17.6), we get, for all f in S(Rn ), WFk ∗ f → LF f 

(17.9)

in S (R ) as k → ∞. Thus, for all f in S(R ), we get, by (17.8) and (17.9), n

n

WF ∗ f  LF f

(17.10)

in the sense of distributions. Thus, by (17.10), the Weyl transform WF ∗ , initially defined on S(Rn ), can be extended to a bounded linear operator from L2 (Rn ) into L2 (Rn ) that is equal to LF . Now, let F ∈ L∞ (Cn ). Then we can find a sequence n {Fk }∞ k1 of simple functions on C such that the Lebesgue measure of the set {z ∈ Cn : Fk (z)  0} is finite for k  1, 2, . . . , and Fk → F a.e. on Cn as k → ∞. Now, for all f and g in S(Rn ), we get, by (12.1), (17.5), and the Lebesgue dominated convergence theorem, (WF ∗ f )(g)  (2π )−n/2 (F ∗ )(z)W (f, g)(z)dz ¯ Cn  lim (2π)−n/2 (Fk ∗ )(z)W (f, g)(z)dz ¯ k→∞

Cn

 lim (WFk ∗ f )(g) k→∞

 lim (LFk f )(g)  (LF f )(g).

(17.11)

WF ∗ f  LF f,

(17.12)

k→∞

So, by (17.11), f ∈ S(Rn ).

17. Compact Localization Operators

85

Therefore, by (17.12), the Weyl transform WF ∗ , initially defined on S(Rn ), can be extended to a bounded linear operator from L2 (Rn ) into L2 (Rn ) that is equal to LF . 2 Theorem 17.2. Let F ∈ Lr (Cn ), 1 ≤ r < ∞. Then the localization operator LF : L2 (Rn ) → L2 (Rn ) is compact. n Proof. Let {Fk }∞ k1 be a sequence of functions in S(C ) such that Fk → F in r n L (C ) as k → ∞. Then, by Theorem 6.8 and Lemma 11.4, LFk  WFk ∗ is a Hilbert–Schmidt operator and hence compact because Fk ∗  is a function in S(Cn ) for k  1, 2, . . . . But by Proposition 15.1 and Theorem 15.4, LFk → LF in B(L2 (Rn )) as k → ∞. Therefore, LF is compact. 2

Remark 17.3. That Theorem 17.2 is false for r  ∞ can be seen easily by taking the function F on Cn to be such that F (z)  1,

z ∈ Cn .

For then, by the resolution of the identity formula, i.e., (15.4), LF is equal to the identity operator on L2 (Rn ) and hence cannot be compact. Remark 17.4. A more precise and more general result than Theorem 17.2 has been proved. See Theorem 6.1 in the paper [15] by He and Wong in this connection.

18 Hermite Polynomials

We are now interested in criteria for the boundedness and/or compactness of Weyl transforms on L2 (R) using an orthonormal basis for L2 (R2 ) consisting of Hermite functions on C ( R2 ), i.e., Wigner transforms of Hermite functions on R. The inverse Fourier transforms of functions in this orthonormal basis, i.e., the Fourier– Wigner transforms of Hermite functions on R, are shown in Chapter 22 to be eigenfunctions of a partial differential operator on C. In this and the next two chapters we lay out the basic properties of Hermite polynomials, Hermite functions, and Laguerre polynomials, which we shall use to study Weyl transforms. A good reference for these topics is Chapter 6 of the book [7] by Folland. The properties of Hermite functions on C to be used in this book are given in Chapters 21, 22, and 23. Let n  0, 1, 2, . . . . Then we define the function Hn on R by  n d 2 n x2 Hn (x)  (−1) e (e−x ), x ∈ R. (18.1) dx We call Hn the Hermite polynomial of degree n. It is easy to see that H0 (x)  1, H1 (x)  2x, and so on. Proposition 18.1. For n  1, 2, . . . ,  Hn (x)  2xHn−1 (x) − Hn−1 (x),

Proof. e

−x 2

x ∈ R.

By (18.1),  Hn (x)  (−1)

n

d dx

n (e

−x 2

 )  (−1)

n

d dx



d dx

n−1

(e−x ) 2

88

18. Hermite Polynomials



d − dx

  n−1  d d 2 n−1 −x 2 (−1) (e )  − (e−x Hn−1 (x)) dx dx

  −(−2xe−x Hn−1 (x) + e−x Hn−1 (x)) 2

2

  2xe−x Hn−1 (x) − e−x Hn−1 (x), 2

2

x ∈ R, 2

and Proposition 18.1 follows.

Remark 18.2. Using Proposition 18.1 repeatedly, we see that the highest power in Hn (x) is equal to (2x)n multiplied by the highest power in H0 (x), which is then equal to 2n x n . Let w be the function on R defined by w(x)  e−x , 2

x ∈ R.

(18.2)

Then we define L2w (R) to be the set of all complex-valued functions on R such that ∞ |f (x)|2 w(x)dx < ∞. −∞

Then L2w (R) is a Hilbert space in which the inner product , w and norm  w are, respectively, given by ∞ f (x)g(x)w(x)dx (18.3) f, g w  −∞

and  f w 

∞

−∞

|f (x)| w(x)dx 2

 21 (18.4)

for all f and g in L2w (R). Proposition 18.3. {Hn : n  0, 1, 2, . . .} is an orthogonal set in L2w (R). Moreover, √ Hn 2w  2n n! π , n  0, 1, 2, . . . . Proof. Let m and n be nonnegative integers such that m ≤ n. Then, by (18.1), (18.2), and (18.3), we get ∞ 2 Hm (x)Hn (x)e−x dx Hm , Hn w  −∞  n ∞ d 2 2 n x2  Hm (x)(−1) e (e−x )e−x dx dx −∞  n ∞ d 2 n  (−1) Hm (x) (e−x )dx dx −∞ ∞ 2  Hm(n) (x)e−x dx. (18.5) −∞

18. Hermite Polynomials

89

Thus, by (18.5), Hm , Hn w  0 if m  n. Moreover, for n  0, 1, 2, . . . , by Remark 18.2, (18.3), (18.4), and (18.5), we get ∞ ∞ √ 2 2 Hn 2w  Hn(n) (x)e−x dx  2n n! e−x dx  2n n! π −∞

−∞

2

and hence complete the proof. We can strengthen Proposition 18.3 and get the following result.

Theorem 18.4. { H1n w Hn : n  0, 1, 2, . . .} is an orthonormal basis for L2w (R). To prove Theorem 18.4, we use two lemmas. Lemma 18.5. Let {pn : n  0, 1, 2, . . .} be a sequence of nonzero polynomials such that the degree of pn is equal to n. Let P be any polynomial of degree k. Then there exist constants c0 , c1 , c2 , . . . , ck such that P 

k


c n pn .

n0

Proof. Let P be a polynomial of degree zero. Then P (x)  α for all x in R, where α is a constant. Suppose that p0 (x)  β for all x in R, where β is a nonzero constant. Then α α P (x)  β  p0 (x), x ∈ R. β β Suppose that Lemma 18.5 is true for all polynomials of degree at most k. Let P be a polynomial of degree k + 1. Then we choose the constant ck+1 such that P and ck+1 pk+1 have the same highest power. Thus, P − ck+1 pk+1 is a polynomial of degree at most k. Hence P − ck+1 pk+1 

k


c n pn

n0

for some constants c0 , c1 , c2 , . . . , ck , and the proof is complete. Lemma 18.6. Let f be a measurable function on R such that ∞ 2 |f (x)|e|xξ | e−x dx < ∞

2

(18.6)

−∞

for all ξ in R. If



∞

−∞

f (x)P (x)e−x dx  0 2

(18.7)

for all polynomials P , then f  0 a.e. on R. Proof. We begin by noting that ∞ ∞
∞ (ixξ )n 2 2 f (x)e−x dx, ξ ∈ R. eixξ f (x)e−x dx  n! −∞ −∞ n0

(18.8)

90

18. Hermite Polynomials

Now,

  ∞ N 

(ixξ )n |xξ |n 2  −x 2  f (x)e  ≤ |f (x)|e−x   n0 n!  n0 n! ≤ e|xξ | |f (x)|e−x , 2

x, ξ ∈ R.

(18.9)

Hence, using (18.6)–(18.9) and the Lebesgue dominated convergence theorem, ∞ ∞
(iξ )n ∞ n 2 ixξ −x 2 e f (x)e dx  x f (x)e−x dx  0. (18.10) n! −∞ −∞ n0 Thus, by (18.10), f (x)e−x complete.

2

 0 for almost all x in R, and the proof is 2

Proof of Theorem 18.4. In view of Proposition 18.3, we need only prove that if f ∈ L2w (R) is such that f, Hn w  0,

n  0, 1, 2, . . . ,

then f  0 a.e. on R. But by the Schwarz inequality,  ∞  21  ∞ |tx| −x 2 2 −x 2 |f (x)|e e dx ≤ |f (x)| e dx −∞

−∞

(18.11)

∞ −∞

e

2|tx| −x 2

e

 21 dx

−1. Then, for n  0, 1, 2, . . . , we define the function Lαn on R by Lαn (x) 

x −α ex n!



d dx

n

(e−x x α+n ),

x > 0.

We call Lαn the Laguerre polynomial of degree n and order α. If we write out Lαn (x), x > 0, in detail, then we get Lαn (x)

 n−k n   n d x −α ex
k −x  (x α+n ), (−1) e n! k0 k dx

x > 0.

(20.1)

Thus, Lαn (x) 

n−1 (α + n)(α + n − 1) · · · (α + k + 1) (−1)n n
x + (−x)k , x > 0. (20.2) n! (n − k)!k! k0

For the highest power in Lαn (x), x > 0, we let k  n in (20.1) or use (20.2) to get Lαn (x) 

(−1)n n x + ···, n!

x > 0.

Let u be the function on (0, ∞) defined by u(x)  x α e−x ,

x ∈ (0, ∞).

(20.3)

96

20. Laguerre Polynomials

Then we define L2u (0, ∞) to be the set of all complex-valued functions f on (0, ∞) such that ∞ |f (x)|2 u(x)dx < ∞. 0

L2u (0, ∞)

Then is a Hilbert space in which the inner product , u and norm  u are, respectively, given by ∞ f, g u  f (x)g(x)u(x)dx 0

and



∞

f u 

|f (x)| u(x)dx 2

 21

0

for all f and g in L2u (0, ∞). Proposition 20.1. {Lαn : n  0, 1, 2, . . .} is an orthogonal set in L2u (0, ∞). Moreover, (α + n + 1) , n  0, 1, 2, . . . . n! Let m and n be nonnegative integers such that m < n. Then ∞ α α Lm , Ln u  Lαm (x)Lαn (x)x α e−x dx 0  n d 1 ∞ α Lm (x) (e−x x α+n )dx  n! 0 dx   (−1)n ∞ d n α  {Lm (x)}e−x x α+n dx  0. n! dx 0 Lαn 2u 

Proof.

Next, by (20.3) and the same computations as before, 1 ∞ −x α+n (α + n + 1) α 2 α α Ln u  Ln , Ln u  e x dx  n! 0 n! for n  1, 2, . . . , and that the same formula is valid for n  0 follows from the definition of Lα0 . 2 Theorem 20.2. { L1α u Lαn : n  0, 1, 2, . . .} is an orthonormal basis for L2u (0, ∞). n

Proof. In view of Proposition 20.1, we only need to prove that if g ∈ L2u (0, ∞) is such that g, Lαn u  0,

n  0, 1, 2, . . . ,

(20.4)

then g  0 a.e. on (0, ∞). Now, for n  0, 1, 2, . . . , we get, by Lemma 18.5, xn 

n
k0

ck Lαk (x),

x > 0,

(20.5)

20. Laguerre Polynomials

97

where c0 , c1 , c2 , . . . , cn are constants. Thus, for n  0, 1, 2, . . . , we get, by (20.4) and (20.5), ∞ ∞ n
n α −x g(x)x x e dx  ck g(x)Lαk (x)x α e−x dx  0. (20.6) 0

0

k0

Let x  y . Then, by (20.6), we get, for n  0, 1, 2, . . . , ∞ ∞ 2 2 2 g(y 2 )y 2n y 2α+1 e−y dy  0 ⇒ 2 g(y 2 )y 2n |y|2α+1 e−y dy  0 0 0 ∞ 2 ⇒ g(y 2 )y 2n |y|2α+1 e−y dy  0. (20.7) 2

−∞

Of course,



∞

−∞

g(y 2 )y 2n+1 |y|2α+1 e−y dy  0, 2

So, by (20.7) and (20.8), ∞ 2 g(y 2 )y n |y|2α+1 e−y dy  0,

n  0, 1, 2, . . . .

n  0, 1, 2, . . . .

−∞

(20.8)

(20.9)

Let P be any polynomial of degree k. Then P (y) 

k


an y n ,

y ∈ R,

n0

where a0 , a1 , a2 , . . . , ak are constants. So, by (20.9), ∞ 2 g(y 2 )P (y)|y|2α+1 e−y dy  0.

(20.10)

−∞

Also, for all ξ in R, ∞ 2 |g(y 2 )||y|2α+1 e|yξ | e−y dy −∞

≤



∞

−∞

 

∞

|g(y )| |y| 2 2

2α+1 −y 2

2 α −x

|g(x)| x e

e

 21  dy

 21  dx

0

∞ −∞

∞ −∞

|y|

|y|

2α+1 2|yξ | −y 2

e

2α+1 −2|yξ | −y 2

e

e

e

 21 dy

 21 dy

< ∞. (20.11)

Thus, by Lemma 18.6, (20.10), and (20.11), g(y 2 )|y|2α+1  0 for almost all y in R. Therefore, g  0 a.e. on (0, ∞). 2 Theorem 20.3. For each fixed positive number x, ∞
n0

e− 1−z  , |z| < 1, (1 − z)α+1 xz

Lαn (x)zn

where the series is uniformly and absolutely convergent on every compact subset of {z ∈ C : |z| < 1}.

98

20. Laguerre Polynomials

e− 1−z the generating function of the Laguerre (1 − z)α+1 α polynomials Ln , n  0, 1, 2, . . . . xz

Remark 20.4. We call

Proof of Theorem 20.3.

γ 

r 

|

|

|

|

|

|

|

|

|

|

|

|

|

|

x

Fig. 3 Let γ be a circle with center at x and lying inside the right half plane (Fig. 3). Now,   ∞ ∞

x −α ex d n −x α+n n Lαn (x)zn  (e x )z n! dx n0 n0 ∞ e−ζ ζ α+n x −α ex
 zn dζ, (20.12) n+1 2π i n0 γ (ζ − x) where the principal branch of ζ α+n is taken, i.e., ζ α+n  e(α+n)Log−π ζ and Log−π ζ  ln |ζ | + iArg−π ζ,

−π < Arg−π ζ < π.

Next, for n  1, 2, . . . ,  −ζ α+n   e ζ  e−Reζ e(α+n) ln |ζ |    (ζ − x)n+1   r n+1 e−(x−r) (x + r)α+n r n+1  α  x+r n −(x−r) (x + r) . e r r ≤

20. Laguerre Polynomials

99

∞
zn e−ζ ζ α+n is uniformly and (ζ − x)n+1 n0 absolutely convergent with respect to z on {z ∈ C : |z| < rx } and ζ on γ , where r rx is any number in (0, x+r ). Therefore, by (20.12),  n −ζ α
∞
zζ e ζ ∞ x −α ex α n Ln (x)z  dζ (20.13) 2π i γ ζ − x n0 ζ − x n0

Then, for all z in C with |z|
0. 1

1

2

Proof of Theorem 23.1. We begin with the formula ∞ 1 2 −x 2 e √ e−u +2ixu du, x ∈ R. π −∞

(23.1)

108

23. Laguerre Formulas for Hermite Functions on C

So, for k  0, 1, 2, . . . ,



 ∞ d k 1 2 e−u +2ixu du √ dx π −∞ ∞ x2 2 1 2 (2iu)k e−u +2ixu du  e− 2 (−1)k ex √ π −∞ ∞ 1 x2 2 uk e−u +2ixu du, x ∈ R.  √ (−2i)k e 2 π −∞ x2

hk (x)  e− 2 (−1)k ex

2

(23.2)

Hence, for any x and y in R and any w in (−1, 1), we have, by (23.1) and (23.2), ∞
hk (x)hk (y)

2k k!

k0

      

wk

∞ 1 1 (x 2 +y 2 )
(−2i)2k ∞ ∞ −u2 −v2 +2ixu+2iyv k k k e2 e u v w du dv π 2k k! −∞ −∞ k0 ∞ ∞ ∞ (−2uvw)k −u2 −v2 +2ixu+2iyv 1 1 (x 2 +y 2 )
e2 e du dv π k! k0 −∞ −∞ 1 1 (x 2 +y 2 ) ∞ ∞ −2uvw −u2 −v2 +2ixu+2iyv e2 e e du dv π −∞ −∞   ∞ 1 1 (x 2 +y 2 ) ∞ −u2 +2ixu 2 e2 e e−v +2iyv−2uvw dv du π −∞ −∞  ∞  1 1 (x 2 +y 2 ) ∞ −u2 +2ixu 2 2 2 2 2 2iyv e2 e e−(v +2uvw+u w )+u w +e dv du π −∞ −∞   ∞ ∞ 1 1 (x 2 +y 2 ) 2 2 2 2 e2 e−u +2ixu+u w e−v e2iy(v−uw) dv du π −∞ −∞ 1 1 (x 2 +y 2 ) −y 2 ∞ −u2 +2ixu+u2 w2 −2iyuw e e du. (23.3) √ e2 π −∞

Therefore, for any x and y in R and any w in (−1, 1), we have, by (23.1) and (23.3), ∞
1 1 (x 2 −y 2 ) ∞ −(1−w2 )u2 +2i(x−yw)u hk (x)hk (y) k 2 w  e du. (23.4) e √ 2k k! π −∞ k0 1

Let (1 − w2 ) 2 u  t. Then, for any x and y in R and any w in (−1, 1), we have, by (23.1) and (23.4), ∞
x−yw hk (x)hk (y) k 1 1 (x 2 −y 2 ) ∞ −t 2 +2i (1−w 1 2 )1/2 t 2 w  e dt(1 − w 2 )− 2 e √ k k! 2 π −∞ k0 1

 e 2 (x

2

2

(x−yw) −y 2 ) − 1−w2

e

 (1 − w 2 )− 2 e 1

− 21

(1 − w 2 )− 2

1+w2 1−w2

1

(x 2 +y 2 )+

2w 1−w2

xy

.

(23.5)

23. Laguerre Formulas for Hermite Functions on C

109

Now, the last term in (23.5) is analytic on the cut plane C−{x ∈ R : x ≤ −1 or x ≥ 1}, and the first term in (23.5) is analytic on the open disk {w ∈ C : |w| < 1}. The two terms are equal on (−1, 1). Hence, by the principle of analytic continuation, the proof of the theorem is complete. 2 Here is a formula expressing the Hermite functions ej,j , j  0, 1, 2, . . . , on C in terms of Laguerre polynomials. Theorem 23.3. For j  0, 1, 2, . . . and any z in C,   1 2 − 21 0 1 2 ej,j (z)  (2π ) Lj |z| e− 4 |z| . 2 Proof. For j  0, 1, 2, . . . , y, p in R, and all r on R with |r| < 1, we get, by Mehler’s formula in Theorem 23.1,   ∞ ∞ 
hk (y + p2 )hk y − p2 k p  p k
ek y + ek y − r  r √ 2 2 2k k! π k0 k0 2 2 2 1 1 − 1 1+r (2y 2 + p )+ 2r (y 2 − p ) 2 4 1−r 2  √ (1 − r 2 )− 2 e 2 1−r 2 π 2 2 2 2 1 1 − 1+r y 2 − 1+r p + 2r y 2 − 2r p 1−r 2 4 1−r 2 1−r 2 4  √ (1 − r 2 )− 2 e 1−r 2 π 2 2 2 1 1 − 1−2r+r y 2 − 1+2r+r p 4 1−r 2  √ (1 − r 2 )− 2 e 1−r 2 π 1 1 1−r 2 1+r p2  √ (1 − r 2 )− 2 e− 1+r y − 1−r 4 . (23.6) π

Taking the inverse Fourier transform of the first and last terms in (23.6) with respect to y, we get, for all z  q + ip in C and all r in R with |r| < 1, ∞ ∞
1 1 1 1+r p2 1−r 2 ek,k (z)r k  √ (1 − r 2 )− 2 e− 1−r 4 √ eiqy e− 1+r y dy π 2π −∞ k0 1 1 − 1 |z|2 r − 1 |z|2 √ e 2 1−r e 4 . (23.7) 2π 1 − r So, by Theorem 20.3 and (23.7),   ∞ 1 2 k − 1 |z|2 1
|z| r e 4 , ek,k (z)r k  √ L0k 2 2π k0 k0

∞


and hence ek,k (z)  (2π )−1/2 L0k for k  0, 1, 2, . . . .



 1 2 − 1 |z|2 |z| e 4 , 2

z ∈ C, 2

Now, we can give another set of formulas expressing Hermite functions on C in terms of Laguerre polynomials.

23. Laguerre Formulas for Hermite Functions on C

110

Theorem 23.4. For j, k  0, 1, 2, . . . and any z in C, j! ej +k,j (z)  (2π )−1/2 { (j +k)! } 2 ( √i 2 )k (¯z)k Lkj ( 21 |z|2 )e− 4 |z| , 1

(i) (ii)

1

2

j! ej,j +k (z)  (2π )−1/2 { (j +k)! } 2 ( √i 2 )k zk Lkj ( 21 |z|2 )e− 4 |z| . 1

1

2

Remark 23.5. Let k  0. Then Theorem 23.4 becomes Theorem 23.3. Remark 23.6. We note that, for j, k  0, 1, 2, . . . , ej,j +k (z)  V (ej , ej +k )(z)  W (ej , ej +k )ˇ(z)  (W (ej +k , ej ))ˇ(z)  (W (ej +k , ej ))ˆ(z)  (W (ej +k , ej ))ˇ(−z)  (V (ej +k , ej ))(−z)  ej +k,j (−z),

z ∈ C.

(23.8)

Thus, if part (i) of Theorem 23.4 is true, then, by (23.8),   21     j! 1 2 − 1 |z|2 i k −1/2 k k ej,j +k (z)  (2π ) |z| e 4 (−z) Lj −√ (j + k)! 2 2     21   j! i k k k 1 2 − 1 |z|2 −1/2 |z| e 4 , z ∈ C.  (2π ) z Lj √ (j + k)! 2 2 Thus, to prove Theorem 23.4, we only need to prove part (i). The following lemma will be used in the proof of Theorem 23.4. Lemma 23.7. For α > −1 and k  1, 2, . . . , d α α+1 (L (x))  −Lk−1 (x), dx k Proof.

x > 0.

By (20.2), k d α (k + α + 1) d
(−x)j (Lk (x))  dx dx j 0 (k − j + 1)(j + α + 1) j !

−

k
j 1

− 

k−1


(−x)j −1 (k + α + 1) (k − j + 1)(j + α + 1) (j − 1)! (k − 1 + α + 1 + 1) (−x)l (k − 1 − l + 1)(l + α + 1 + 1) l!

l0 −Lα+1 k−1 (x),

2

x > 0.

Proof of Theorem 23.4. In view of Remark 23.6, it is enough to prove part (i). By Theorem 23.3, the formula is true if k  0. Suppose that the formula is true for all nonnegative integers j and all nonnegative integers k with k ≤ l, say. Then, by part (i) of Theorem 22.1, ej +k+1,j  −i(2j + 2)− 2 Zej +k+1,j +1 . 1

(23.9)

23. Laguerre Formulas for Hermite Functions on C

111

Now, by the induction hypothesis, we have, for all z ∈ C,      21  (j + 1)! i k k k 1 2 − 1 |z|2 ej +k+1,j +1 (z)  (2π )−1/2 |z| e 4 . (¯z) Lj +1 √ (j + k + 1)! 2 2 (23.10) Let fj be the function on C defined by   1 2 − 1 |z|2 k k |z| e 4 , z ∈ C. fj (z)  (¯z) Lj +1 (23.11) 2 Then, for k ≥ 1,

       ∂fj 1 2 1 2 1 1 1 2 2 (z)  (¯z)k (∂Lkj +1 ) |z| qe− 4 |z| + Lkj +1 |z| − q e− 4 |z| ∂q 2 2 2   1 1 2 + k(¯z)k−1 Lkj +1 |z|2 e− 4 |z| , z ∈ C, (23.12) 2

and        ∂fj 1 2 1 2 i 1 1 2 2 i (z)  (¯z)k (∂Lkj +1 ) |z| ipe− 4 |z| + Lkj +1 |z| − p e− 4 |z| ∂p 2 2 2   1 2 − 1 |z|2 + k(¯z)k−1 Lkj +1 |z| e 4 , z ∈ C. (23.13) 2 So, by (23.11), (23.12), and (23.13), (Zfj )(z)  (¯z)k+1 (∂Lkj +1 )



 1 2 − 1 |z|2 |z| e 4 , 2

z ∈ C.

(23.14)

It is easy to see that (23.14) is also true for k  0. Thus, by (23.9)–(23.11) and (23.14), ej +k+1,j (z)



 21 (j + 1)!  (2π ) (−i)(2j + 2) (j + k + 1)!    k i 1 2 − 1 |z|2 |z| e 4 × √ (¯z)k+1 (∂Lkj +1 ) 2 2 −1/2

− 21

(23.15) for all z in C. But, by (23.15) and Lemma 23.7, ej +k+1,j (z)

   21   (j + 1)! i k k+1 k+1 1 2 − 1 |z|2 |z| e 4 (¯z) Lj √ (j + k + 1)! 2 2     21  k+1 j! i 1 2 − 1 |z|2 |z| e 4 , z ∈ C,  (2π )−1/2 (¯z)k+1 Lk+1 √ j (j + k + 1)! 2 2 and the proof is complete. 2  (2π)−1/2 i(2j + 2)− 2

1



24 Weyl Transforms on L2(R) with Radial Symbols

For Weyl transforms on L2 (R) with radial symbols, we can give a sufficient and necessary condition for boundedness. A sufficient and necessary condition for compactness can also be given. In order to obtain these conditions, we need the Wigner transforms of Hermite functions on R. For j, k  0, 1, 2, . . . , we define the function ψj,k on R2 by ψj,k (x, ξ )  W (ej , ek )(x, ξ ),

x, ξ ∈ R.

Theorem 24.1. For j, k  0, 1, 2, . . . , we get, for any ζ  x + iξ ,   21 √ 1 2 j! (i) ψj +k,j (ζ )  2(−1)j (2π)− 2 (j +k)! ( 2)k (ζ¯ )k Lkj (2|ζ |2 )e−|ζ | ,   21 √ 1 2 j! ( 2)k ζ k Lkj (2|ζ |2 )e−|ζ | . (ii) ψj,j +k (ζ )  2(−1)j (2π)− 2 (j +k)! Proof. It is easy to see from Proposition 18.1 that ek is even or odd if k is, respectively, even or odd. Thus, for k  0, 1, 2, . . . , e˜k (x)  ek (−x)  (−1)k ek (x), So, by (24.1), we get V (ej , e˜k )(q, p)  (2π)−1/2



x ∈ R.

(24.1)

 p  p e˜k y − dy eiqy ej y + 2 2 −∞ ∞

 (−1)k V (ej , ek )(q, p),

q, p ∈ R.

(24.2)

Thus, by Lemma 14.4 and (24.2), W (ej , ek )(x, ξ )  (−1)k 2V (ej , ek )(−2ξ, 2x)  (−1)k 2ej,k (−2ξ, 2x), x, ξ ∈ R.

(24.3)

114

24. Weyl Transforms on L2 (R) with Radial Symbols

For all ζ  x + iξ in C, we get, by (24.3), ψj,k (ζ )  (−1)k 2ej,k (2iζ ).

(24.4)

Hence, by Theorem 23.4 and (24.4), ψj +k,j (ζ )  2(−1)j ej +k,j (2iζ )    21  j! i k k 2 j −1/2 2 (−i)k (ζ¯ )k Lkj (2|ζ |2 )e−|ζ |  2(−1) (2π ) √ (j + k)! 2   21 √ j! 2  2(−1)j (2π )−1/2 ( 2)k (ζ¯ )k Lkj (2|ζ |2 )e−|ζ | , ζ ∈ C, (j + k)! and ψj,j +k (ζ )  2(−1)j +k ej,j +k (2iζ )   21   j! i k k k k k 2 j +k −1/2 2 i ζ Lj (2|ζ |2 )e−|ζ |  2(−1) (2π ) √ (j + k)! 2   21 √ j! 2 2  2(−1)j (2π )−1/2 ( 2)k ζ k Lkj (2|ζ |2 )e−|ζ | , ζ ∈ C. (j + k)! To see the role of the function ψj,k , j, k  0, 1, 2, . . . , in the study of the Weyl transform, we use the following result in the paper [24] by Simon. We skip the proof. Theorem 24.2. Let f be any function in S(R). Then f 

∞


f, ek ek ,

k0

where the convergence is in S(R), i.e., for all nonnegative integers α and β,   N  
 α β  α β sup x (∂ f )(x) − f, ek x (∂ ek )(x) → 0  x∈R  k0 as N → ∞. Let σ be a tempered function on R2 . Suppose that σ is radial, i.e., σ (x, ξ )  σ (r), x, ξ ∈ R,  where r  x 2 + ξ 2 . Now, by Theorem 24.1, for j, k  0, 1, 2, . . . and j ≥ k,   21 √ k! 1 2 j −k ψj,k (x, ξ )  2(−1)k (2π )− 2 ( 2)j −k (ζ¯ )j −k Lk (2|ζ |2 )e−|ζ | (24.5) j! for all ζ  x + iξ in C. Now, for all f and g in S(R), we can use (12.1), Theorem 24.2, the bilinearity of W , the proof of Theorem 12.1, and (24.5) to get (Wσ f )(g) ¯  (2π )−1/2 σ (W (f, g))

24. Weyl Transforms on L2 (R) with Radial Symbols

  (2π)

−1/2

−1/2

 (2π)



σ W f, σ

 ∞


∞


115

 g, ek ek

k0



g, ¯ ek W (f, ek )

k0

 (2π)−1/2  (2π)−1/2

∞


g, ¯ ek σ (W (f, ek ))

k0 ∞ ∞



g, ¯ ek f, ej σ (ψj,k ).

(24.6)

k0 j 0

Remark 24.3. We have shown only that (24.6) is valid in the sense that we sum with respect to j first and then with respect to k. Now, for j, k  0, 1, 2, . . . and j ≥ k, we get, by (24.5), ∞ ∞ σ (x, ξ )ψj,k (x, ξ )dx dξ σ (ψj,k )  −∞



2π

−∞



∞

k

σ (ρ)2(−1) (2π) 0

− 21



0

k! j!

 21

√ j −k ( 2)j −k ρ j −k e−i(j −k)θ Lk (2ρ 2 )

× e−ρ ρdρdθ   21 ∞ 2π k! 1 1 e−i(j −k)θ dθ σ (ρ) 2 2 (j −k)+1 (−1)k (2π )− 2 ρ j −k+1  j ! 0 0 2

j −k

× Lk

(2ρ 2 )e−ρ dρ. 2

(24.7)

Thus, by (24.7), σ (ψj,k )  0,

j  k.

(24.8)

So, by (24.6), (24.7), and (24.8), we get ¯  (2π )−1/2 (Wσ f )(g)

∞


f, ek g, ¯ ek σ (ψk,k ),

(24.9)

k0

where 1



σ (ψk,k )  (2π ) 2 (−1)k 2 0

∞

σ (ρ)L0k (2ρ 2 )e−ρ ρdρ, 2

k  0, 1, 2, . . . , (24.10)

for all f and g in S(R). Remark 24.4. The convergence in (24.9) is valid in the sense that the sequence of partial sums of the series is convergent. Theorem 24.5. Let σ be a tempered function on R2 . Suppose that σ is radial, i.e., σ (x, ξ )  σ (ρ),

x, ξ ∈ R,

116

24. Weyl Transforms on L2 (R) with Radial Symbols

where ρ 



x 2 + ξ 2 . For k  0, 1, 2, . . . , let ∞ 2 σ (ρ)L0k (2ρ 2 )e−ρ ρdρ. ak  0

Then Wσ is a bounded linear operator from L2 (R) into L2 (R) if and only if the sequence {ak }∞ k0 is bounded, (ii) Wσ is a compact operator from L2 (R) into L2 (R) if and only if ak → 0 as k → ∞. (i)

Proof.

Suppose that there is a positive constant M such that |ak | ≤ M,

k  0, 1, 2, . . . .

(24.11)

Then, for all f and g in S(R), we get, by (24.9) and (24.10), (Wσ f )(g)  (2π )−1/2

∞


1

(2π) 2 (−1)k 2ak f, ek g, ek ,

(24.12)

k0

and hence, by the Schwarz inequality and (24.11),  21   21  ∞ ∞

|(Wσ f )(g)| ≤ 2M | f, ek |2 | g, ek |2 k0

k0

 2Mf L2 (R) gL2 (R) . Thus, Wσ f L2 (R) ≤ 2Mf L2 (R) ,

f ∈ S(R).

Therefore, by a limiting argument, Wσ is a bounded linear operator from L2 (R) into L2 (R). Conversely, suppose that Wσ is a bounded linear operator from L2 (R) into L2 (R). Then, by (24.12), (Wσ ej )(ej )  (−1)j 2aj ,

j  0, 1, 2, . . . .

(24.13)

Therefore, by (24.13) and the Schwarz inequality, |aj |  |(Wσ ej )(ej )|  | Wσ ej , ej | ≤ Wσ ej L2 (R) ej L2 (R) ≤ Wσ ∗ ,

j  0, 1, 2, . . . ,

and part (i) is proved. To prove part (ii), suppose that ak → 0 as k → ∞. For N  0, 1, 2, . . . , we define the bounded linear operator WN : L2 (R) → L2 (R) by WN f, g 

N


(−1)k 2ak f, ek g, ¯ ek

k0

for all f and g in L2 (R). Then WN f 

N
k0

(−1)k 2ak f, ek ek ,

f ∈ L2 (R),

(24.14)

24. Weyl Transforms on L2 (R) with Radial Symbols

117

for N  0, 1, 2, . . . . Therefore, WN : L2 (R) → L2 (R) is a finite rank operator, and hence compact for N  0, 1, 2, . . . . Now, for all f and g in S(R), by (24.12) and (24.14), | (Wσ − WN )f, g | ≤ 2( sup |ak |)f L2 (R) gL2 (R) k≥N +1

(24.15)

for N  0, 1, 2, . . . . Then, by (24.15), Wσ − WN ∗ ≤ 2( sup |ak |) → 0 k≥N +1

as N → ∞. Therefore, Wσ is the limit in B(L2 (R)) of a sequence of compact operators from L2 (R) into L2 (R). So, Wσ is a compact operator from L2 (R) into L2 (R). Conversely, suppose that Wσ is a compact operator from L2 (R) into L2 (R). Since ej → 0 weakly in L2 (R) as j → ∞, it follows that Wσ ej → 0 in L2 (R) as j → ∞. But by (24.13) and the Schwarz inequality, 2|aj | ≤ Wσ ej L2 (R) ej L2 (R)  Wσ ej L2 (R) → 0 as j → ∞.

2

25 Another Fourier Transform

We compute in this chapter the Fourier transform of a function related to the Laguerre polynomial of degree k and order 0, k  0, 1, 2, . . . . This Fourier transform will be used in the next chapter to obtain a criterion for the compactness of the Weyl transform on L2 (R). We begin with some complex analysis. Let f be a continuous function on [0, ∞) such that we can find a positive constant A and a constant c for which |f (t)| ≤ Aect ,

t ≥ 0.

Then we define the function F on the region {z ∈ C : Rez > c} by ∞ F (z)  e−zt f (t)dt, Rez > c.

(25.1)

(25.2)

0

The function F is in fact the Laplace transform of f . Theorem 25.1. F is analytic on the region {z ∈ C : Rez > c}. To prove Theorem 25.1, we need some preparations. Lemma 25.2. Let ϕ be an entire function. Let g be a continuous function on a closed and bounded interval [a, b] and let G be the function on C defined by b G(z)  g(t)ϕ(zt)dt, z ∈ C. (25.3) a

Then G is continuous on C. Proof. Let z0 ∈ C. Let {zk }∞ k1 be a sequence of complex numbers such that zk → z0 as k → ∞. Let D be a fixed disk centered at z0 such that z0 t ∈ D for all t

120

25. Another Fourier Transform

in [a, b]. Then ϕ is uniformly continuous on D. So, for any given positive number ε, there exists a positive number δ such that |ϕ(z) − ϕ(w)| < ε for all z and w in D with |z − w| < δ. Thus, there exists a positive integer K such that k ≥ K ⇒ |ϕ(zk t) − ϕ(z0 t)| < ε for all t in [a, b]. So, g(t)ϕ(zk t) → g(t)ϕ(z0 t) uniformly with respect to t on [a, b] as k → ∞. Therefore, G(zk ) → G(z0 ) as k → ∞. 2 Lemma 25.3. The function G on C defined by (25.3) is entire. Proof. Let C be a simple closed curve in C. Then   b G(z)dz  g(t)ϕ(zt)dt dz C C a    b b  g(t)ϕ(zt)dz dt  g(t) ϕ(zt)dz dt  a

C

a

C

 0. So, by Lemma 25.2 and Morera’s theorem, G is entire.

2

Proof of Theorem 25.1. We see that by (25.1) and (25.2), F (z) exists for all z in the region {z ∈ C : Rez > c}. Indeed, ∞ ∞ −zt |e f (t)|dt  e−(Rez)t |f (t)|dt 0 0 ∞ A ≤A < ∞. e−(Rez−c)t dt  Rez −c 0 Next, for j  1, 2, . . . , we define the function Fj on C by j Fj (z)  e−zt f (t)dt, z ∈ C. (25.4) 0

By Lemma 25.3, Fj is entire for j  1, 2, . . . . Now, let c1 > c. Then, for Rez > c1 , we get, by (25.1) and (25.4), j j e−(Rez−c)t dt ≤ A e−(c1 −c)t dt |Fj (z) − Fl (z)| ≤ l l  −(c1 −c)l  e − e−(c1 −c)j A →0 c1 − c as j, l → ∞. Thus, Fj → F uniformly on the region {z ∈ C : Rez > c1 } as j → ∞, and hence F is analytic on {z ∈ C : Rez > c1 }. But c1 is an arbitrary number larger than c. So, F is analytic on the region {z ∈ C : Rez > c}. 2

25. Another Fourier Transform

121

Corollary 25.4. For k  0, 1, 2, . . . , the function Fk defined on the region {z ∈ C : Rez > 0} by ∞ Fk (z)  e−zt t k dt, Rez > 0, 0

is analytic. Proof. Let ε be any positive number. Then we can find a positive constant Aε such that |t k | ≤ Aε eεt ,

t ≥ 0.

So, by Theorem 25.1, Fk is analytic on the region {z ∈ C : Rez > ε}. Since ε is an arbitrary positive number, it follows that Fk is analytic on {z ∈ C : Rez > 0}. 2 For k  0, 1, 2, . . . , we define the function lk on R by  x −2 0 lk (x)  e Lk (x), x > 0, 0, x ≤ 0.

(25.5)

The aim of this chapter is to compute the Fourier transform of the function lk , k  0, 1, 2, . . . . Theorem 25.5. For k  0, 1, 2, . . . ,   k 1 ˆlk (ξ )  2 2iξ − 1 , π 2iξ + 1 2iξ + 1 Proof.

ξ ∈ R.

By (25.5) and the definition of Lk (x), x > 0, ∞ 1 lˆk (ξ )  (2π )− 2 e−ixξ lk (x)dx 0

  ex d k −x k  (2π ) e e (e x )dx k! dx 0 ∞  k d (−1)k 1 x (2π)− 2 (e 2 (1−2iξ ) )e−x x k dx  k! dx 0  k ∞ k (−1) x − 21 (1−2iξ ) 1 − 2iξ 2 (2π) e e−x x k dx  k! 2 0   ∞ (−1)k 1 − 2iξ k 1 x (2π)− 2 e− 2 (1+2iξ ) x k dx  k! 2 0 − 21



∞

−ixξ − x2

(25.6)

for all ξ in R. Now, for z > 0,  k ∞ ∞ 1 s ds k! −zt k −s  k+1 (k + 1)  k+1 . e t dt  e (25.7) z z z z 0 0

∞ By Corollary 25.4, 0 e−zt t k dt is analytic on {z ∈ C : Rez > 0}. So, by (25.7) and the principle of analytic continuation, ∞ k! e−zt t k dt  k+1 , Rez > 0. (25.8) z 0

122

25. Another Fourier Transform

So, by (25.8), we get ∞

e−x(

1+2iξ 2

0

x dx 

) k

k! ( 1+2iξ )k+1 2

,

ξ ∈ R.

(25.9)

Therefore, by (25.6) and (25.9),

  2k+1 1 − 2iξ k 1 lˆk (ξ )  (2π)− 2 (−1)k 2 (1 + 2iξ )k+1   k 2 2iξ − 1 1 , ξ ∈ R.  π 2iξ + 1 2iξ + 1

2

26 A Class of Compact Weyl Transforms on L2(R)

The space Lr∗ (R), 1 ≤ r < ∞, defined by (14.1) will be used again in this chapter to give a criterion for the compactness of the Weyl transform on L2 (R). Let g ∈ Lr∗ (R), 1 ≤ r < ∞, and let σ be the function on R2 defined by 

where ρ  ∞ −∞

σ (x, ξ )  g(ρ ˆ 2 ), +

r

x, ξ ∈ R,

Then σ ∈ L (R ). Indeed, 2π ∞ ∞   |σ (x, ξ )|r dx dξ  |g(ρ ˆ 2 )|r ρdρdθ −∞ 0 0 ∞  r π |g(ρ)| ˆ dρ ≤ π g ˆ rLr  (R2 ) < ∞. x2

ξ 2.

2

0

Theorem 26.1. Let g ∈ defined by where ρ 



Lr∗ (R),

1 ≤ r < ∞, and let σ be the function on R2

σ (x, ξ )  g(ρ ˆ 2 ),

x, ξ ∈ R,

x 2 + ξ 2 . Then Wσ : L2 (R) → L2 (R) is a compact operator.

Proof. By part (ii) of Theorem 24.5, we only need to prove that the sequence {ak }∞ k1 of complex numbers defined by ∞ 2 ak  σ (ρ)L0k (2ρ 2 )e−ρ ρdρ, k  0, 1, 2, . . . , 0

where σ (ρ)  g(ρ ˆ ), is such that limk→∞ ak  0. To do this, let lk and Ik be functions on R defined, respectively, by (25.5) and 2

Ik (x)  lk (2x),

x ∈ R,

(26.1)

26. A Class of Compact Weyl Transforms on L2 (R)

124

for k  0, 1, 2, . . . . Then, by Theorem 25.5 and (26.1),     iξ − 1 k 1 1 ξ 1 Iˆk (ξ )  lˆk , √ 2 2 2π iξ + 1 iξ + 1

ξ ∈ R,

for k  0, 1, 2, . . . . Now, for 1 < s < ∞, we get, by (26.2),  1s  ∞ 1 1 ˆ Ik Ls (R)  √ dξ s 2π −∞ |iξ + 1|

(26.2)

(26.3)

and 1 Iˆk L∞ (R) ≤ √ 2π

(26.4)

for k  0, 1, 2, . . . . Thus, for 1 < s ≤ ∞, by (26.3) and (26.4) we can get a positive constant Cs such that Iˆk Ls (R) ≤ Cs ,

k  0, 1, 2, . . . .

Note that for k  0, 1, 2, . . . , ∞ ∞ 2 2 σ (ρ)L0k (2ρ 2 )e−ρ ρdρ  g(ρ ˆ 2 )L0k (2ρ 2 )e−ρ ρdρ ak  0 0∞ 1 1 ∞ 0 −t g(t)L ˆ g(t)I ˆ  k (t)dt. k (2t)e dt  2 0 2 −∞ Now, if



∞

−∞

g(t)I ˆ k (t)dt 

∞

−∞

g(t)Iˆk (t)dt, k  0, 1, 2, . . . ,

then, by (26.2), (26.6), and (26.7),   ∞ iξ − 1 k 1 1 ak  √ dξ, g(ξ ) iξ + 1 iξ + 1 2 2π −∞ Let ξ  tan θ, − π2 < θ < 1 ak  √ 2 2π



π 2

− π2

k  0, 1, 2, . . . .

(26.5)

(26.6)

(26.7)

(26.8)

Then, by (26.8), we get, for k  0, 1, 2, . . . ,   i tan θ − 1 k 1 sec2 θ dθ. g(tan θ) (26.9) i tan θ + 1 i tan θ + 1 π . 2

But for k  0, 1, 2, . . . and − π2 < θ < π2 ,   i tan θ − 1 k (−1)k (cos θ − i sin θ)k   (−1)k e−2ikθ . i tan θ + 1 (cos θ + i sin θ)k Thus, for k  0, 1, 2, . . . , we have, by (26.9) and (26.10), (−1)k π ak  √ f (θ)e−2ikθ dθ, 2 2π −π

(26.10)

(26.11)

26. A Class of Compact Weyl Transforms on L2 (R)

where f (θ) 

⎧ ⎨ 0, ⎩

2

g(tan θ) sec θ , i tan θ+1

0,

125

−π < θ < − π2 , − π2 < θ < π2 , π < θ < π. 2

Note that f ∈ L1 [−π, π]

(26.12)

because by H¨older’s inequality there exists a positive constant Cr such that   π  ∞  g(ξ )  2  g(tan θ) sec2 θ     dθ   dξ  i tan θ + 1    − π2 −∞ iξ + 1  ∞  1 r 1 ≤ gLr (R) dξ < ∞.  r −∞ |iξ + 1| Thus, by (26.11), (26.12), and the Riemann–Lebesgue lemma, ak → 0 as k → ∞. Thus, the proof is complete, provided that we can prove (26.7). To do this, we first note that for 1 ≤ r ≤ 2, (26.7) follows from Lemma 14.6. For 2 < r < ∞, by ˆ ˆ (26.5) we can find a sequence {ψj }∞ j 1 of functions in S(R) such that ψj → Ik in ∞ r L (R) as j → ∞. Now, let {ϕl }l1 be a sequence of functions in S(R) such that ϕl → g in Lr (R) as l → ∞. Then ϕl → g in S  (R) as l → ∞, so that ϕˆl → gˆ in S  (R) as l → ∞. Therefore, ∞ ∞ g(t)Iˆk (t)dt  lim ϕl (t)Iˆk (t)dt l→∞ −∞ −∞ ∞  lim lim ϕl (t)ψˆ j (t)dt l→∞ j →∞ −∞ ∞  lim lim ϕˆl (t)ψj (t)dt. (26.13) l→∞ j →∞ −∞

Now, if we can prove that ∞ lim lim ϕˆl (t)ψj (t)dt  lim lim l→∞ j →∞ −∞

∞

j →∞ l→∞ −∞

then, by (26.13) and (26.14), ∞ ˆ g(t)Ik (t)dt  lim −∞

∞

j →∞ −∞

ϕˆl (t)ψj (t)dt,

g(t)ψ ˆ j (t)dt.

(26.14)

(26.15)

But by the Fourier inversion formula and the Hausdorff–Young inequality, ψj → Ik in Lr (R) as j → ∞. So, by (26.15), ∞ ∞ g(t)Iˆk (t)dt  g(t)I ˆ k (t)dt. −∞

−∞

So, it remains to prove (26.14), or equivalently, ∞ ˆ lim lim ϕl (t)ψj (t)dt  lim lim l→∞ j →∞ −∞

∞

j →∞ l→∞ −∞

ϕl (t)ψˆ j (t)dt.

(26.16)

126

26. A Class of Compact Weyl Transforms on L2 (R)

To prove (26.16), let ε be any positive number. Then we pick N1 to be any positive integer such that ε j ≥ N1 ⇒ ψˆ j − Iˆk Lr  (R) < , (26.17) 2M1 where M1  sup{ϕl Lr (R) : l  0, 1, 2, . . .}.

(26.18)

Let N2 be another positive integer such that ε

l ≥ N2 ⇒ ϕl − gLr (R)
j. So, by (27.1)–(27.4),



σ (ψj,k )  aj k

π −π

(27.3) (27.4)

(θ)e−i(j −k)θ dθ

ˆ − k),  aj k (2π)(j

j, k  0, ±1, ±2, . . . ,

(27.5)

27. A Class of Bounded Weyl Transforms on L2 (R)

where ˆ (l) 



1 2π

π

−π

(θ)e−ilθ dθ,

l  0, ±1, ±2, . . . .

129

(27.6)

Let f and g be in S(R). Then, by (24.6) and (27.5), there exists a positive constant C  such that   ∞ ∞

   |(Wσ f )(g)|  (2π )−1/2  g, ek f, ej σ (ψj,k )  k0 j 0  ≤ C

∞ ∞



ˆ − k) f, ej  g, ek | |(j

k0 j 0

 C

∞


| g, ek |

≤ C

ˆ − j ) f, ej | |(k

j 0

k0



∞


∞


 21 ⎛  2 ⎞ 21 ∞ ∞

ˆ − j ) f, ej | ⎠ | g, ek |2 ⎝ |(k

k0

k0

⎛  C  gL2 (R) ⎝

 ∞ ∞



j 0

2 ⎞ 21 ˆ − j ) f, ej | ⎠ , |(k

(27.7)

j 0

k0

where ˜ (θ)  (θ), −π ≤ θ ≤ π.

(27.8)

By Lemma 27.2, (27.6), and (27.8), ∞


ˆ |(k)| < ∞.

(27.9)

k−∞

So, by (27.7), (27.8), (27.9), and Young’s inequality,   21  ∞ ∞

 2 ˆ |(k)| | f, ek | |(Wσ f )(g)| ≤ C gL2 (R)   C

k−∞ ∞




k0

ˆ |(k)| f L2 (R) gL2 (R)

(27.10)

k−∞

for all f and g in S(R). Therefore, by (27.10),   ∞
 ˆ |(k)| f L2 (R) , Wσ f L2 (R) ≤ C

f ∈ S(R),

k−∞

and the proof is complete.

2

28 A Weyl Transform with Symbol in S (R2)

We can compute in this chapter the eigenvalues and eigenfunctions of a specific Weyl transform with symbol in S  (R2 ) as a compact and self-adjoint operator on L2 (R). Let ρ be a positive real number. Then we define the linear functional δρ : S(R2 ) → C by π δρ (ϕ)  ϕ(ρeiθ )ρdθ, ϕ ∈ S(R2 ). −π

Then it is easy to check that δρ is a tempered distribution on R2 . For k  0, 1, 2, . . . , we define the number λk by λk  2(−1)k ρL0k (2ρ 2 )e−ρ . 2

(28.1)

Theorem 28.1. Wδρ : L2 (R) → L2 (R) is a compact and self-adjoint operator. Furthermore, the nonzero eigenvalues of Wδρ : L2 (R) → L2 (R) are precisely equal to the numbers defined by (28.1), and ∞


|λk |r < ∞,

r > 4.

k0

Proof.

Let f and g be in S(R). Then, by (24.6), (Wδρ f )(g)  (2π )−1/2

∞ ∞

g, ek f, ej δρ (ψj,k ). k0 j 0

(28.2)

132

28. A Weyl Transform with Symbol in S  (R2 )

For j, k  0, 1, 2, . . . , by (27.3) and (27.4), we get δρ (ψj,k ) ⎧

π 1 √ 1 2 j −k ⎪ ⎪ 2(−1)k (2π )− 2 jk!! 2 ( 2)j −k −π ρ j −k+1 e−i(j −k)θ Lk (2ρ 2 )e−ρ dθ, ⎪ ⎨ j ≥ k, 

π 1 √ 1 2 k−j j ⎪ ⎪ 2(−1) (2π )− 2 jk!! 2 ( 2)k−j −π ρ k−j +1 e−i(j −k)θ Lj (2ρ 2 )e−ρ dθ, ⎪ ⎩ k ≥ j. (28.3) Thus, by (28.3), δρ (ψj,k ) 



0, j  k, 1 2 2(−1)k (2π) 2 ρL0k (2ρ 2 )e−ρ , j  k.

(28.4) (28.5)

So, for all f and g in S(R), by (28.1), (28.2), (28.4), and (28.5), (Wδρ f )(g) 

∞


λk f, ek g, ek .

(28.6)

k0

We now use an asymptotic formula for the Laguerre polynomials in the book [28] by Szeg˝o, which states that for arbitrary but fixed positive numbers ε and ω we get  π 1 x 1 1 1 3 (28.7) + O(k − 4 ), x > 0, L0k (x)  π − 2 e 2 x − 4 k − 4 cos 2(kx) 2 − 4 as k → ∞, where the O-term is uniform with respect to x on [ε, ω]. Thus, by (28.1) and (28.7), there exists a positive constant C such that |λk | ≤ Ck − 4

1

(28.8)

for k large enough, say, for k ≥ k0 . Therefore, for r > 4, by (28.8), we get ∞


|λk |r ≤ C r

kk0

∞


k − 4 < ∞. r

(28.9)

kk0

Now, for every positive integer N , the linear operator WN : L2 (R) → L2 (R) defined by WN f 

N


λk f, ek ek , f ∈ L2 (R),

(28.10)

k0

is of finite rank and hence compact. Moreover, for all f and g in S(R), by (28.6) and (28.10),   ∞ 
   λk f, ek g, ek  |(Wδρ f )(g) − (WN f )(g)| ≤  kN +1  ≤ ( sup |λk |)f L2 (R) gL2 (R) . k≥N +1

Thus, by (28.9) and (28.11), Wδρ − WN ∗ ≤ sup |λk | → 0 k≥N +1

(28.11)

28. A Weyl Transform with Symbol in S  (R2 )

133

as N → ∞. Therefore, Wδρ : L2 (R) → L2 (R) is compact. To see that Wδρ : L2 (R) → L2 (R) is self-adjoint, we note that for all f and g in S(R), by (28.6), Wδρ f, g  (Wδρ f )(g) ¯ 

∞


λk f, ek g, ¯ ek

k0

and f, Wδρ g  Wδρ g, f  (Wδρ g)(f¯) 

∞


λk f, ek g, ¯ ek .

k0

For k  0, 1, 2, . . . , λk is an eigenvalue of Wδρ : L2 (R) → L2 (R) because for any g in S(R), by (28.6), we get (Wδρ ej )(g) 

∞


λk ej , ek g, ek  (λj ej )(g),

j  0, 1, 2, . . . .

k0

Therefore, λj is an eigenvalue of Wδρ : L2 (R) → L2 (R) for j  0, 1, 2, . . . . Finally, let λ be an eigenvalue of Wδρ : L2 (R) → L2 (R) and let f ∈ L2 (R) be a corresponding eigenfunction. Then   ∞ ∞

Wδρ f, ej ej  λ f, ej ej j 0

⇒

∞


j 0

f, ej Wδρ ej 

j 0

⇒

∞
j 0

∞


f, ej λej

j 0

f, ej λj ej 

∞


f, ej λej

j 0

⇒ f, ej λj  f, ej λ,

j  0, 1, 2, . . . .

(28.12)

Since {ej : j  0, 1, 2, . . .} is an orthonormal basis for L2 (R) and f  0, it follows that f, ek  0 for some k  0, 1, 2, . . . . Thus, by (28.12), λ  λk . So, every eigenvalue of Wδρ : L2 (R) → L2 (R) is equal to λk for some k  0, 1, 2, . . . . 2 To put Theorem 28.1 in a proper perspective of quantum mechanics, it is imperative to note that due to quantization described at the end of Chapter 4, the numerical values of measurements of an observable in classical mechanics with phase space R2n are replaced by the spectrum of the self-adjoint Weyl transform representing the observable. This explains why the computation of the spectrum of a self-adjoint Weyl transform on L2 (Rn ) is an important chapter in applied mathematics.

29 The Symplectic Group

It is shown in this chapter that the Weyl transform is invariant with respect to the symplectic group, i.e., the group of all symplectic linear transformations from Cn into Cn . A linear transformation a : Cn → Cn satisfying [a(z), a(z )]  [z, z ],

z, z ∈ Cn ,

(29.1)

where [ , ] is the symplectic form on C defined by (8.1), is said to be symplectic. We provide a sufficiently self-contained treatment of symplectic linear transformations from Cn into Cn and the symplectic group in this chapter. Related matters can be found in the books [9], [13], and [34] by Greub, Halmos, and Weyl, respectively. n

Remark 29.1. Let a : Cn → Cn be a linear transformation. We call the matrix of a with respect to the standard basis for Cn ( R2n ) the standard matrix of a. We n shall q identify a with its standard matrix. A point (q, p) in C will also be denoted by p . Proposition 29.2. Let a : Cn → Cn be  a linear transformation such that the A B standard matrix of a is equal to , where A, B, C, and D are n × n C D matrices with real entries. Then a is symplectic if and only if At D − C t B  I, At C  C t A,

(29.2) (29.3)

B t D  D t B,

(29.4)

and where ( ) is the transpose of the matrix ( ) and I is the n × n identity matrix. t

136

29. The Symplectic Group

Proof. For j  1, 2, . . . , n, we let εj be the point in Rn such that all coordinates except the j th coordinate are equal to zero, and the j th coordinate is equal to 1. Then, for j, k  1, 2, . . . , n,      εj Aεj A B a(εj , 0)  , (29.5)  C D 0 Cεj  a(0, εk ) 

A C

B D

    Bεk 0  , εk Dεk

(29.6)

and hence, by (8.1), (29.5), and (29.6), [a(εj , 0), a(0, εk )]  2(εkt B t Cεj − εkt D t Aεj ).

(29.7)

Since by (8.1), [(εj , 0), (0, εk )]  −2εkt εj ,

j, k  1, 2, . . . , n,

(29.8)

it follows from (29.1), (29.7), and (29.8) that D t A−B t C  I , and (29.2) is proved. Next, for j, k  1, 2, . . . , n, we get, by (8.1) and (29.5), [a(εj , 0), a(εk , 0)]  2(εkt At Cεj − εkt C t Aεj )

(29.9)

[(εj , 0), (εk , 0)]  0.

(29.10)

and So, by (29.1), (29.9), and (29.10), (29.3) is proved. Finally, for j, k  1, 2, . . . , n, we get, by (8.1) and (29.6), [a(0, εj ), a(0, εk )]  2(εkt B t Dεj − εkt D t Bεj )

(29.11)

[(0, εj ), (0, εk )]  0.

(29.12)

and So, by (29.1), (29.11), and (29.12), (29.4) is proved. Conversely, for z  x + iξ and z  x  + iξ  in Cn , we get      x Ax + Bξ A B a(z)   (29.13) C D ξ Cx + Dξ and a(z ) 



A C

B D

      Ax + Bξ  x  . ξ Cx  + Dξ 

(29.14)

Hence, by (8.1), (29.13), and (29.14), [a(z), a(z )]  2(x  )t (At C − C t A)x + 2(ξ  )t (B t D − D t B)ξ + 2(x  )t (At D − C t B)ξ − 2x t (At D − C t B)ξ  . (29.15) So, by (8.1), (29.2), (29.3), (29.4), and (29.15), [a(z), a(z )]  2(x  · ξ − x · ξ  )  [z, z ], and hence a is symplectic.

2

29. The Symplectic Group

137

Proposition 29.3. Let Sp(n, R) be the set of all symplectic linear transformations from Cn into Cn . Then Sp(n, R) is a group with respect to the usual composition of mappings.     A1 B 1 A2 B 2 Proof. Let and be in Sp(n, R). Then C1 D 1 C 2 D2      A 2 B2 A B A1 B1  , C1 D 1 C2 D2 C D where A  A1 A2 + B1 C2 , B  A1 B2 + B1 D2 , C  C1 A2 + D1 C2 , and D  C1 B2 + D1 D2 . So, some easy computations and Proposition 29.2 give At D − C t B  I, At C  C t A, and B t D  D t B, and hence, by Proposition 29.2 again,   A2 A1 B1 C 1 D1 C2

B2 D2

 ∈ Sp(n, R).

The associative law follows  fromthe usual association law for the compositions I 0 of mappings. The matrix in Sp(n, R) is obviously the identity element.  0 I A B Finally, let a  ∈ Sp(n, R). Let C D   t −B t D . b −C t At Then, by Proposition 29.2,     t  −B t A B I 0 D  . C D −C t At 0 I   A B So, is invertible, and C D   −1  t −B t D A B  . −C t At C D

138

29. The Symplectic Group

Therefore, it remains to prove that b ∈ Sp(n, R). To do this, we note that for all z and z in Cn , we get, by (29.1), [z, z ]  [(ab)(z), (ab)(z )]  [a(b(z)), a(b(z ))]  [b(z), b(z )], i.e., b is symplectic. Corollary 29.4. Let Proof.



A C

B D

2



∈ Sp(n, R). Then AB t  BAt and CD t  DC t .

From the proof of Proposition 29.3,  −1  t  A B −B t D ∈ Sp(n, R).  C D −C t At

Hence, by Proposition 29.2, the proof is complete.

2

Proposition 29.5. Let R be the irreducible and unitary representation of the Heisenberg group H n on L2 (Rn ) defined by (8.5). Let a ∈ Sp(n, R). Then the mapping Raλ : H n → G defined by λ

Raλ (z, t)  R λ (a(z), t), (z, t) ∈ H n ,

(29.16)

is also an irreducible and unitary representation of H n on L2 (Rn ). Proof.

Let (z, t) and (z , t  ) be in H n . Then, by (8.2) and (29.1), Raλ ((z, t) · (z , t  ))  Raλ (z + z , t + t  + [z, z ])  R λ (a(z + z ), t + t  + [z, z ])  R λ (a(z) + a(z ), t + t  + [a(z), a(z )])  R λ ((a(z), t) · (a(z ), t  ))  R λ (a(z), t)R λ (a(z ), t  )  Raλ (z, t)Raλ (z , t  ).

Next, it is easy to see that for all f in L2 (Rn ), we get, by (29.16), Raλ (z, t)f  R λ (a(z), t)f → f in L2 (Rn ) as (z, t) → (0, 0). Thus, Raλ : H n → G is a unitary representation of H n on L2 (Rn ). Finally, let M be a closed subspace of L2 (Rn ) such that M is invariant under all the operators Raλ (z, t), (z, t) ∈ H n . Let (z, t) ∈ H n and let w ∈ Cn be such that z  a(w). Then M is invariant under the unitary operator Raλ (w, t). But by (29.16), Raλ (w, t)  R λ (a(w), t)  R λ (z, t). So, M is invariant under all the operators R λ (z, t), (z, t) ∈ H n , and using the irreducibility of the unitary representation R λ of H n on L2 (Rn ), we conclude that M  L2 (Rn ) or M  {0}. Therefore, Raλ is irreducible. 2 Proposition 29.6. The irreducible and unitary representations Raλ and R λ of H n on L2 (Rn ) are equivalent for all a in Sp(n, R).

29. The Symplectic Group

Proof.

139

We first note that by (8.5), 1

Raλ (0, t)  R λ (a(0), t)  R λ (0, t)  e 4 iλt ,

t ∈ R. 2

So, by Proposition 8.6 and Remark 8.7, the proof is complete.

Remark 29.7. Let λ  1. Then, for all a in Sp(n, R), we get, by Proposition 29.6, a unitary operator Va on L2 (Rn ) such that R 1 (a(z), t)  Ra1 (z, t)  Va R 1 (z, t)Va−1 , (z, t) ∈ H n , and hence, by Remark 8.8, ρ(a(z))  R 1 (a(z), 0)  Ra1 (z, 0)  Va ρ(z)Va−1 ,

z ∈ Cn .

Proposition 29.8. Let a ∈ Sp(n, R). Then there exists a unitary operator Ua on L2 (Rn ) such that ρ(a(z)) ˜  Ua ρ(z)Ua−1 ,

z ∈ Cn ,

where a˜  (a t )−1 . Proof. In view of Remark 29.7, we only need to prove that a˜ ∈ Sp(n, R). In view of Proposition 29.3, we only need to prove thatthe transpose of any a in  A B Sp(n, R) is in Sp(n, R). To this end, let a  ∈ Sp(n, R) and let C D   0 I J  . (29.17) −I 0 Then, by Proposition 29.2, it is easy to see that J ∈ Sp(n, R). It has been shown in the proof of Proposition 29.3 that   0 −I (29.18) J −1  I 0 and a −1 



Dt −C t

So, by (29.17), (29.18), and (29.19),  t    t A Ct 0 −I D t a   I 0 B t Dt −C t

−B t At

 .

−B t At



(29.19)

0 −I

and hence a t ∈ Sp(n, R).

I 0



 J −1 a −1 J, 2

Remark 29.9. For each a in Sp(n, R), the unitary operator Ua on L2 (Rn ), generated by Proposition 29.8, is uniquely determined up to a constant multiple in the sense that if Ua and Va are unitary operators on L2 (Rn ) such that ρ(a(z)) ˜  Ua ρ(z)Ua−1 ,

z ∈ Cn ,

(29.20)

ρ(a(z)) ˜  Va ρ(z)Va−1 ,

z ∈ Cn ,

(29.21)

and

140

29. The Symplectic Group

then Va  ca Ua for some constant ca . Indeed, by (29.20) and (29.21), we get Ua ρ(z)Ua−1  Va ρ(z)Va−1 ,

z ∈ Cn ,

ρ(z)Ua−1 Va  Ua−1 Va ρ(z),

z ∈ Cn .

and hence (29.22)

So, by (8.5) and (29.22), R 1 (z, t)Ua−1 Va  Ua−1 Va R 1 (z, t), (z, t) ∈ H n . n

1

(29.23) 2

n

Since R is an irreducible and unitary representation of H on L (R ), it follows that the only bounded linear operators on L2 (Rn ) that commute with all operators R 1 (z, t), (z, t) ∈ H n , are constant multiples of the identity I on L2 (Rn ). Hence, by (29.23), we get a constant ca such that Ua−1 Va  ca I, or Va  ca Ua . Proposition 29.10.   0 I (i) Let a  . Then Ua  F.  −I 0 I 0 (ii) Let a  , where C is a symmetric matrix. Then C I 1

(Ua f )(x)  e 2 i(Cx)·x f (x),

x ∈ Rn ,

for all f inL2 (Rn ).  A 0 (iii) Let a  , where A is an invertible matrix. Then 0 (At )−1 (Ua f )(x)  f (A−1 x)| det A|− 2 , 1

x ∈ Rn ,

for all f in L2 (Rn ). Proof.

For all z in Cn and all f in S(Rn ), we get, by (2.1),

(ρ(z)fˇ)(x)  eiq·x+ 2 iq·p fˇ(x + p) 1  eiq·x+ 2 iq·p (2π)−n/2 ei(x+p)·ξ f (ξ )dξ Rn iq·x+ 21 iq·p −n/2 e (2π) ei(x+p)·(η−q) f (η − q)dη Rn 1  eiq·x+ 2 iq·p (2π)−n/2 eix·η−ix·q+ip·η−iq·p f (η − q)dη Rn 1 −n/2  (2π) eix·η eip·η− 2 iq·p f (η − q)dη 1

Rn

 (ρ(p, −q)f )ˇ(x),

x ∈ Rn .

(29.24)

29. The Symplectic Group

Also,

 a(q, ˜ p) 

0 −I

I 0

    q p  , p −q

q, p ∈ Rn .

141

(29.25)

So, by (29.24) and (29.25), ρ(a(z)) ˜  Fρ(z)F −1 ,

z ∈ Rn ,

and hence Ua  F. To prove part (ii), we first note that  −1   I C I −C t −1 a˜  (a )   , 0 I 0 I and hence, for all z  q + ip in Cn ,      q q − Cp I −C a(z) ˜   . 0 I p p

(29.26)

Now, we only need to prove that the Ua given by the equation in part (ii) satisfies ˜ )(x), (Ua ρ(z)Ua−1 f )(x)  (ρ(a(z))f

x ∈ Rn ,

(29.27)

for all z in Cn and all f in L2 (Rn ). But, by (29.26), (Ua ρ(z)Ua−1 f )(x)  e 2 i(Cx)·x (ρ(z)Ua−1 f )(x) 1

 e 2 i(Cx)·x eiq·x e 2 iq·p e− 2 i(C(x+p))·(x+p) f (x + p) 1

1

1

1

1

1

1

 e 2 i(Cx)·x+iq·x+ 2 iq·p− 2 i(Cx)·x−i(Cx)·p− 2 i(Cp)·p f (x + p) 1

 ei(q−Cp)·x+ 2 i(q−Cp)·p f (x + p)  (ρ(q − Cp, p)f )(x)  (ρ(a(z))f ˜ )(x),

x ∈ Rn ,

for all z in Cn and all f in L2 (Rn ). Thus, (29.27) follows. To prove part (iii), we note that −1  t −1   t 0 0 (A ) A  , a˜  0 A−1 0 A and hence, for all z  q + ip in Cn ,  t −1     t −1  q (A ) q (A ) 0 a(z) ˜   . 0 A p Ap

(29.28)

We only need to prove that the Ua given by the equation in part (iii) satisfies ˜ )(x), (Ua ρ(z)Ua−1 f )(x)  (ρ(a(z))f

x ∈ Rn ,

(29.29)

for all z in Cn and all f in L2 (Rn ). But (Ua−1 f )(x)  f (Ax)| det A| 2 , 1

x ∈ Rn ,

(29.30)

142

29. The Symplectic Group

and hence, by (29.28) and (29.30), (Ua ρ(z)Ua−1 f )(x)  (ρ(z)Ua−1 f )(A−1 x)| det A|− 2

1

 eiq·(A

−1

x)

t −1

 ei((A ) e

e 2 iq·p (Ua−1 f )(A−1 x + p)| det A|− 2 1

1

q)·x+ 21 iq·p

f (x + Ap)

i((At )−1 q)·x+ 21 i((At )−1 q)·(Ap)

f (x + Ap)

t −1

 (ρ((A ) q, Ap)f )(x)  (ρ(a(z))f ˜ )(x),

x ∈ Rn , 2

for all z in Cn and all f in L2 (Rn ). Thus, (29.29) is proved.

Remark 29.11. It can be proved that the group Sp(n, R) is finitely generated by matrices of the three types given in Proposition 29.10. So, we are able to compute Ua , up to a constant multiple, for all a in Sp(n, R). It can also be proved that the constants can be chosen such that Ua Ub  ±Uab ,

a, b ∈ Sp(n, R).

Thus, the mapping from Sp(n, R) into G given by a → Ua is a “metaplectic representation” of Sp(n, R) on L2 (Rn ). See Chapter 4 of the book [6] by Folland for a discussion of the metaplectic representation and related matters. Remark 29.12. The unitary operators Ua provided by Proposition 29.10 are obviously homeomorphisms from S(Rn ) onto S(Rn ). It is also easy to see that the unitary operators Ua of the types given by parts (i) and (ii) of Proposition 29.10 are homeomorphisms from S  (Rn ) onto S  (Rn ). Let Ua be a unitary operator of the type given by part (iii) of Proposition 29.10. Then, for all f in S  (Rn ), we define Ua f : S(Rn ) → C by (Ua f )(ϕ)  f (Ua−1 ϕ),

ϕ ∈ S(Rn ).

It is then easy to check that Ua is a homeomorphism from S  (Rn ) onto S  (Rn ). Let σ ∈ S  (R2n ) and a ∈ Sp(n, R). Then we define the mapping σ ◦ a : S(R2n ) → C by (σ ◦ a)(ϕ)  σ (ϕ ◦ a −1 ),

ϕ ∈ S(R2n ).

(29.31)

We leave it as an exercise to prove that σ ◦ a ∈ S  (R2n ). We can now give the main result in this chapter. Theorem 29.13. Let a ∈ Sp(n, R). Then, for all σ in S  (R2n ), Wσ ◦a  Ua−1 Wσ Ua , where Ua is any unitary operator guaranteed by Proposition 29.8. Proof. Let σ ∈ S(R2n ). Then, for all ϕ and ψ in S(Rn ), we get, by (2.3), (3.12), Remark 29.12, Theorem 3.1, Theorem 4.3, and the adjoint formula for the Fourier

29. The Symplectic Group

143

transform, Ua−1 Wσ Ua ϕ, ψ  Wσ Ua ϕ, Ua ψ σ (x, ξ )W (Ua ϕ, Ua ψ)(x, ξ )dx dξ  (2π)−n/2 n C  (2π)−n/2 σˆ (z)V (Ua ϕ, Ua ψ)(z)dz n C  (2π)−n σˆ (z) ρ(z)Ua ϕ, Ua ψ dz n C σˆ (z) Ua−1 ρ(z)Ua ϕ, ψ dz. (29.32)  (2π )−n Cn

So, by (29.32), Proposition 29.8, and Remark 29.11, Ua−1 Wσ Ua ϕ, ψ  (2π)−n σˆ (z) ρ(a t (z))ϕ, ψ dz Cn −n  (2π) σˆ (a(z)) ρ(z)ϕ, ˜ ψ dz n C  (2π)−n (σ ◦ a)ˆ(z) ρ(z)ϕ, ψ dz Cn

 Wσ ◦a ϕ, ψ for all ϕ and ψ in S(Rn ). So, Wσ ◦a  Ua−1 Wσ Ua ,

σ ∈ S(R2n ).

(29.33)

Let σ ∈ S  (R2n ). Then, by Proposition 1.18, we can find a sequence {σj }∞ j 1 of functions in S(R2n ) such that σj → σ in S  (R2n ) as j → ∞. Then, by (29.31), (29.33), and Remark 29.12, we get, for all ϕ in S(Rn ), Wσj ◦a ϕ → Wσ ◦a ϕ and Ua−1 Wσj Ua ϕ → Ua−1 Wσ Ua ϕ in S  (Rn ) as j → ∞, and hence Wσ ◦a  Ua−1 Wσ Ua .

2

30 Symplectic Invariance of Weyl Transforms

We make precise in this chapter the fact that the symplectic invariance property in Theorem 29.13 characterizes the quantization σ → σ (x, D), described at the end of Chapter 4, as the Weyl transform Wσ . The main result, i.e., Theorem 30.1, obtained by Shale in [23], is another testimonial to the vision of Hermann Weyl that Wσ is the correct quantization of the classical observable σ . We begin with a proposition. Proposition 30.1. Let σ be the function on R2n defined by σ (x, ξ )  ei

n

j 1

xj

x, ξ ∈ Rn .

,

(30.1)

Then, for all q  (q1 , q2 , . . . , qn ) and p  (p1 , p2 , . . . , pn ) in Rn with qj  0, j  1, 2, . . . , n, there exists an element a in Sp(n, R) such that (σ ◦ a)(x, ξ )  σ (a(x, ξ ))  ei(q·x+p·ξ ) , Proof.

x, ξ ∈ Rn .

(30.2)

By (30.1), σ (x, ξ )  ei(u,0)·(x,ξ ) ,

x, ξ ∈ Rn ,

where u is the point in R in which all coordinates are equal to 1. Let   Q P a , 0 Q−1

(30.3)

n

where

⎛

q1 ⎜0 Q⎝

0 q2

0

0

⎞ ··· 0 ··· 0 ⎟ ⎠ ··· · · · qn

(30.4)

146

30. Symplectic Invariance of Weyl Transforms

and

⎛

p1 ⎜ 0 P ⎝

0 p2

0

0

⎞ ··· 0 ··· 0 ⎟ ⎠. ··· · · · pn

By Proposition 29.2, a ∈ Sp(n, R). Now, by (30.4),      x Qx + P ξ Q P  a(x, ξ )  , 0 Q−1 ξ Q−1 ξ

x, ξ ∈ Rn .

(30.5)

Thus, by (30.3) and (30.5), (σ ◦ a)(x, ξ )  ei(u,0)·(Qx+P ξ )  ei(q·x+p·ξ ) ,

x, ξ ∈ Rn , 2

and the proposition is proved.

Let L(S(Rn ), S  (Rn )) be the set of all continuous linear mappings from S(Rn ) n  n into S  (Rn ). Let {Ak }∞ k1 be a sequence of elements in L(S(R ), S (R )) and let A n  n n  n be in L(S(R ), S (R )). We say that Ak → A in L(S(R ), S (R )) as k → ∞ if Ak ϕ → Aϕ 

in S (R ) as k → ∞ for all ϕ in S(Rn ). We can now state and prove the main result in the last chapter of the book. n

Theorem 30.2. Let A : S  (R2n ) → L(S(Rn ), S  (Rn )) be a linear mapping. We suppose that it is continuous in the sense that σk → σ in S  (R2n ) ⇒ Aσk → Aσ in L(S(Rn ), S  (Rn )) as k → ∞. Moreover, we suppose that ((Aσ )ϕ)(x)  σ (x)ϕ(x), ∞

x ∈ Rn ,

(30.6)

for all ϕ in S(R ) and all σ in L (R ), and n

n

A(σ ◦ a)  Ua−1 (Aσ )Ua

(30.7)

for all σ in S  (R2n ) and all a in Sp(n, R), where Ua is a unitary operator on L2 (Rn ) given by Proposition 29.8. Then Aσ  Wσ ,

σ ∈ S  (R2n ).

To prove Theorem 30.2, we need a lemma. Lemma 30.3. For any q and p in Rn , let eq,p be the function on R2n such that eq,p (x, ξ )  ei(q·x+p·ξ ) ,

x, ξ ∈ Rn .

Then Weq,p  ρ(q, p), where ρ(q, p) is given by (2.1).

q, p ∈ Rn ,

30. Symplectic Invariance of Weyl Transforms

Proof. we get

147

Let q and p be in Rn . Then, by (2.3), (3.12), Theorem 3.1, and (12.1), ¯  (2π )−n/2 (Weq,p ϕ)(ψ)



Rn

Rn

ei(q·x+p·ξ ) W (ϕ, ψ)(x, ξ )dx dξ

 (2π )n/2 W (ϕ, ψ)ˇ(q, p)  (2π)n/2 V (ϕ, ψ)(q, p)  ρ(q, p)ϕ, ψ ,

ϕ, ψ ∈ S(Rn ), 2

and hence the proof is complete.

Proof of Theorem 30.2. We begin with the observation that by Theorem 29.13, (30.1), (30.2), (30.6), (30.7), and Lemma 30.3, Aeq,p  Weq,p  ρ(q, p)

(30.8)

for all q and p in Rn such that every coordinate in q is nonzero. Let {Qk }∞ k1 be a sequence of concentric cubes in Cn with centers at the origin and edges parallel to the coordinate axes, and limk→∞ |Qk |  ∞, where |Qk | is the volume of Qk , k  1, 2, . . . . Then, using (9.4) and the Lebesgue dominated convergence theorem, we get, for all σ in S(R2n ) and all ϕ in S(Rn ), −n (ρ(q, p)ϕ)(x)σˆ (q, p)dq dp, x ∈ Rn . (30.9) (Wσ ϕ)(x)  lim (2π ) k→∞

Qk

Let σ ∈ S(R2n ). Then, by (30.9), Fubini’s theorem, and the Lebesgue dominated convergence theorem, −n (2π) (ρ(q, p)ϕ)(·)σˆ (q, p)dq dp → Wσ ϕ (30.10) Qk 

in S (R ) as k → ∞ for all ϕ in S(Rn ). Now, for k  1, 2, . . . , we let σk be the function on R2n defined by σk (x, ξ )  (2π )−n eq,p (x, ξ )σˆ (q, p)dq dp, x, ξ ∈ Rn . n

Qk

Then σk → σ in S  (R2n ). Indeed, using Fubini’s theorem and the Fourier inversion formula, we get   −n σk (ψ)  (2π ) σˆ (q, p) eq,p (x, ξ )ψ(x, ξ )dx dξ dq dp Qk Cn ˇ σˆ (q, p)ψ(q, p)dq dp, ψ ∈ S(R2n ), (30.11)  Qk

and hence, by the Lebesgue dominated convergence theorem and the adjoint formula for the Fourier transform, σk (ψ) → σ (ψ)

148

30. Symplectic Invariance of Weyl Transforms

as k → ∞ for all ψ in S(R2n ). So, by continuity, (Aσk )(ϕ) → (Aσ )(ϕ) in S  (Rn ) as k → ∞ for all ϕ in S(Rn ). Now, by (30.8), (2π)−n (ρ(q, p)ϕ)(x)σˆ (q, p)dq dp  lim Sl (x), l→∞

Qk

(30.12)

x ∈ Rn ,

(30.13)

where Sl (x) is a Riemann sum of the form
Sl (x)  (2π)−n (ρ(qj , pj )ϕ)(x)σˆ (qj , pj )δj  ((Asl )ϕ)(x),

x ∈ Rn , (30.14) where (qj , pj ) is a point in Qk chosen in such a way that each coordinate in qj is nonzero;
sl (x, ξ )  (2π )−n eqj ,pj (x, ξ )σˆ (qj , pj )δj , x, ξ ∈ Rn , (30.15)

and sl (x, ξ ) → (2π )

−n

eq,p (x, ξ )σˆ (q, p)dq dp, x, ξ ∈ Rn , Qk

as l → ∞. But by (30.11) and (30.15), we get  
sl (ψ)  (2π )−n σˆ (qj , pj ) eqj ,pj (x, ξ )ψ(x, ξ )dx dξ δj Cn
ˇ j , pj )δj  σˆ (qj , pj )ψ(q ˇ → σˆ (q, p)ψ(q, p)dq dp  σk (ψ) (30.16) Qk

as l → ∞ for all ψ in S(R2n ). Thus, by (30.16), sl → σk in S  (R2n ) as l → ∞, and hence, by continuity, (Asl )(ϕ) → (Aσk )(ϕ)

(30.17)

in S  (Rn ) as l → ∞ for all ϕ in S(Rn ). Next, for all ψ in S(Rn ), we get, by (2.3), (30.14), and Fubini’s theorem,  
Sl (ψ)  (2π )−n σˆ (qj , pj ) (ρ(qj , pj )ϕ)(x)ψ(x)dx δj Rn   → (2π )−n σˆ (q, p) (ρ(q, p)ϕ)(x)ψ(x)dx dq dp Q Rn k (ρ(q, p)ϕ)(·)σˆ (q, q)dq dp)(ψ)  ((2π )−n Qk

as l → ∞, and hence Sl → (2π)−n

(ρ(q, p)ϕ)(·)σˆ (q, q)dq dp Qk

(30.18)

30. Symplectic Invariance of Weyl Transforms

in S  (Rn ) as l → ∞. So, by (30.14), (30.17), and (30.18), (Aσk )(ϕ)  (2π)−n (ρ(q, p)ϕ)(·)σˆ (q, q)dq dp.

149

(30.19)

Qk

Thus, by (30.10), (30.12), and (30.19), (Aσ )(ϕ)  Wσ ϕ,

ϕ ∈ S(Rn ),

and hence Aσ  Wσ ,

σ ∈ S(R2n ).

(30.20)

Finally, let σ ∈ S (R ). Then, by Proposition 1.18, there is a sequence {σj }∞ j 1 of functions in S(R2n ) such that σj → σ in S  (R2n ) as j → ∞. So, for all ϕ and ψ in S(Rn ), we get, by (12.1) and (30.20), 

2n

¯ (Wσ ϕ)(ψ)  (2π )−n/2 σ (W (ϕ, ψ)) ¯  lim (2π)−n/2 σj (W (ϕ, ψ)) j →∞

 lim (Wσj ϕ)(ψ) j →∞

 lim ((Aσj )ϕ)(ψ), j →∞

and hence (Aσj )(ϕ) → Wσ ϕ 

n



n

(30.21)

in S (R ) as j → ∞. But by continuity, (Aσj )(ϕ) → (Aσ )(ϕ)

(30.22)

in S (R ) as j → ∞. So, by (30.21) and (30.22), Aσ  Wσ , and the proof is complete. 2

References

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Notation Index

B(L2 (Rn )), 22

L(S(Rn ), S  (Rn )), 146

C0∞ , 2

R λ , 39 Raλ , 138

ej,k , 101 en , 94 fˆ, 4 fˇ, 6 Ff , 4 F −1 f , 6 f ∗ g, 3 f ∗λ g, 43 f ⊗ g, 29 G, 39 H n , 10 Hn , 87 hn , 93 lk , 121 LF , 76 Lαn , 95 Lr∗ , 71

S, 2 Sh , 25 Sp(n, R), 137 Tσ , 8 V (f, g), 10 Wσ , 19 W (f ), 13 W (f, g), 15 (W H )n , 75 ∂α, 2 ρ(q, p), 9 σ (x, D), 24 ψj,k , 113 [z, w], 37  ∗ , 22  H S , 35

Index

Adjoint formula, 5, 44, 142, 147 admissible wavelet, 76 annihilation operator, 93, 103, 106 Bracket operation, 38 Canonical commutation relations, 38 classical mechanics, 23, 133 commutator, 38 convolution, 2, 3 creation operator, 93, 103, 106 Differential operator, 1, 2 dilation operator, 5 Dirac delta, vi, 62 Eigenfunction, vi, 93, 106, 131, 133 eigenvalue, vi, 93, 106, 131, 133 Fourier inversion formula, 5, 8, 62, 68, 125, 147 Fourier series, 127, 128 Fourier transform, vi, 4–6, 13, 30, 44, 71, 72, 79, 81, 83, 119, 121 Fourier–Wigner transform, 9, 10, 13, 16, 101 functional calculus, 24, 25, 33

H ∗ -algebra, 33–35 harmonic oscillator, 93, 103, 106 Hausdorff–Young inequality, 71–73, 125 Heisenberg group, vi, 37, 38, 138 Hermite function, vi, 87, 93, 101, 103, 106, 107, 109 Hermite polynomial, 87, 90 Hilbert–Schmidt operator, 25, 26, 32, 34, 57, 58, 74, 85 Inverse Fourier transform, 6, 30, 83, 109 irreducible representation, 40, 42, 75, 76, 138, 140 Joint probability distribution, 13 Kernel, 25, 26, 32, 57 Laguerre polynomial, 87, 95, 98, 107, 109, 132 Laplace transform, 119 Laplacian, 106 left-invariant vector field, 38 Lie algebra, 38 localization operator, vi, 77–79, 83, 85

158

Index

Mehler’s formula, 107, 109 metaplectic representation, 142 modulation operator, 5 momentum, 1, 13, 23 Moyal identity, 15, 16, 101 multiplication operator, 23 Nondeterministic statistical dynamics, 15 Observable, 1, 24, 35, 133, 145 Partial differential operator, 2, 8, 20, 87, 106 phase space, 23, 77, 133 Plancherel’s theorem, 6, 15, 46, 71, 101 Planck’s constant, 1, 23 position, 13, 23 projective representation, 10 pseudo-differential operator, v, vi, 7, 8 Quantization, vi, 19, 23, 35, 133, 145 quantum mechanics, v, 15, 24, 37, 38, 133 Radial symbol, 113–115 regularization, 3 representation, vi, 19, 76, 142 resolution of the identity, 76, 77, 85 Riemann–Lebesgue lemma, 4, 125 Riesz–Thorin theorem, 47, 49, 57, 71, 78 Schr¨odinger representation, 10, 39 signal analysis, 75, 77

spectrum, 133 square-integrable representation, vi, 76 state, 13 Stone–von Neumann theorem, 42 structure equations, 38 symbol, vi, 2, 8, 19–21, 29, 33, 37, 43, 55, 59, 75, 77, 83, 113–115 symplectic form, 37, 38, 43, 135 symplectic group, vi, 135, 137 symplectic invariance, 145 Tempered distribution, vi, 6, 7, 59, 61, 131 tempered function, 7, 114, 115, 128 tensor product, 29, 57 three lines theorem, 47 translation operator, 5 twisted convolution, vi, 37, 43, 44 twisting operator, 30 Unitarily equivalent representations, 42 unitary representation, 39, 40, 42, 44, 75, 76, 138, 140 Vector field, 103 Weyl calculus, 33 Weyl–Heisenberg group, 75 Weyl transform, v, vi, 7, 9, 19, 21, 29, 32, 35, 37, 43, 44, 55, 59, 63, 71, 83–85, 87, 113, 114, 123, 127, 128, 131, 145 Wigner transform, vi, 9, 13, 15, 21, 59 Young’s inequality, 3, 129

Universitext

(continued)

Rotman: Galois Theory Rubel/CoIIiander: Entire and Meromorphic Functions Sagan: Space-Filling Curves Samelson: Notes on Lie Algebras Schiff: Normal Families Shapiro: Composition Operators and Classical Function Theory Simonnet: Measures and Probability Smith: Power Series From a Computational Point of View Smorynski: Self-Reference and Modal Logic Stillwell: Geometry of Surfaces Stroock: An Introduction to the Theory of Large Deviations Sunder: An Invitation to von Neumann Algebras Tondeur: Foliations on Riemannian Manifolds Wong: Weyl Transforms Zong: Strange Phenomena in Convex and Discrete Geometry