A basis in the space of solutions of a convolution equation

A BASIS IN THE SPACE OF SOLUTIONS OF A CONVOLUTION EQUATION V. V. Napalkov

i. Introduction. Let D be an arbitrary convex region in the complex plane C; H(D) be the space of analytic functions in D with the topology of compact convergence; G n be an in0 0 creasing sequence of convex compacts in D exhausting D, and G n c Gn+ I (n = i, 2 . . . . ) (G n is the set of interior points of the compact Gn). We introduce functions hn(8) = k n ( ~ ) , 0 < O ~ 2~ (n = i, 2 . . . . ), where kn(O) = sup Re (z exp iO) is the support function of the compact G n. It is known (see, e.g., [i, z~

n

pp. 77 and 96]) that the Laplace transform establishes a topological isomorphism between the strong dual H*(D) [of the space H(D)] and the space of the entire functions P = lim m ind Pm, where

Let F ~ equation

H*(D).

A functional F determines

in the space H(D) the homogeneous

convolution

31v [y] (z) - - (F, Y (z + t)) = O. If r

is the characteristic

(1)

function of Eq. (i), i.e~

r (~) = L (F) (X) - - (F, exp (Xz)). then by (11, m I) . . . . . (lj, mj) . . . . we denote the sequence A = {lj}j=1 ~ of zeros of the function #(I) with respective multiplicities mj. By W we denote the set of all solutions of Eq. (i) in the space H(D). Then the system

l i ! j ~mt-I l zVexp@iz)

E= b e l o n g s t o W.

L e t G be t h e c o n j u g a t e d i a g r a m [2] o f t h e f u n c t i o n

r

and D be t h e maximal

subregion of D that is covered by all possible shifts G~ of the compact G through vectors =, provided G~ c D. The following result is well known in the theory of convolution equations [3, p. 357]. if zeros {lj} of the functiQn r constitute a regular set,% then the system {exp(ljz)} is a basis in the space of solutions of Eq. (i) analytic in D; it can be shown that this system is also a Schauder basis. And if the function ~(I) has a completely regular growth, then

m]--1

~- W we can uniquely assign a series ~j=1 ~v=0 to every function f(z)-=

.ej~zPexp (~jz) whose par-

tial sums (arranged in a certain way) converge to f(z) in the region D; the system {zP exp (Xjz)} will already not be a basis in the general case. This system serves the purpose of proving that, with the proviso that ~(I) is a function of completely regular growth, in the corresponding space W there always exists a Schauder basis composed of linear combinations of elements of the system E. To construct this basis, the results of [4] are used; in doing *A set {bk} is called regular if (I) the set {bk} has a finite density (limr_~(n(r)/r) < =, n(r) is the number of points {bk} in the disc Ill < r); (2) for some constant c there exists a finite limit: '

(3) for some ~ > 0 the inequality

-

- - I b ~ . J < r

'

'

Ibk+ll - tbkl > ~.

Bashkir Branch of the Academy of Sciences of the USSR. Translated from Matematicheskie Zametki, Vol. 43, No. i, pp. 44-55, January, 1988. Original article submitted March 3, 1987.

-0001-4346/88/4312-0027

$12.50

9

1988 Plenum Publishing

Corporation

27

so, it turns out that the condition of "slow decrease" of the function r (which has been used in [4]) is, within the framework of the present article, naturally related to the con-' dition of completely regular growth of r 2. Preliminary Results. In what follows, it will be assumed that a function r with an indicatrix of growth h(O) has a completely regular growth and a region D coincides with the union U=~QG=, where Q is some convex region and G= is the shift by a vector = of the compact G [the conjugate diagram of the function ~(X)]. It follows from the condition of completely regular growth that outside some set of discs S = {Sj},

Sj={X~C:

I~--ajl ]o'

contained

in a disc Sj the inequality

Therefore ,

I

I

and we finally obtain

+

expI--(2>1:0

Thus, the set of discs S* has zero linear density. Now construct [6, p. 79] an entire function w(X) with the indicator h(O) whose zero {bk} lies outside the discs S* and is regular. Outside the discs I~ - bkl < exp(-Ibkl~0), ~ < D0 < i, for the function w(%) the uniform #A set of discs {i ~ C:

I~ - ~jl < rj} (j = i, 2 . . . . ) has zero linear density if

lira,. -~,(I/r),..i!, ~

28

jl ~) dz YJ (~) -- ,,: :v is)(t~-- ~) defines an entire function of I. This observation and the fact that the region C \~ 6jJj contains (see [6, p. 43]) annular regions {X: rj < l i i < rj+1}, rj + ~, imply uniform con-

29

convergence of series (4) on any compact of the complex plane. Consequently, series (4) defines an entire function g(X). The function g(l) belongs to P. Indeed, consider the discs

B s. = {s where 0 < ~ < i, X j e ~

z~lj.

Let first

Sj;

a point

Ixj~

X lie

[)~--ai IJo,

s

c=const,

we obtain 1

i=_~,1 .~exp[--cl&lq , Taking into account all the inequalities

Iz-Xl where c I is some constant.

X~Rj,

l>/o,

z~lj,

The last inequality and (5) give the estimate

~j(z)

z-~lj, ~ B j .

Ig(X)l ~ B e x p [hN ( a r g X ) e < hj, (arg X) for some obtained estimate of the plane. Therefore, g ( X ) ~ tion that Jp(V)(X k) = 0, residues one can directy fore

z~lj.

obtained above, we finally have

~exp(--c*l~]l~)'

~(z)(~ -z)

where A i s a c o n s t a n t ,

~,~o~Rj,

< A e x P ( - - c , l ;.l~)" exp ( - - % [ z l), 8 0 > 0 ' Substitute

(6)

in (4);

for the function

(6) g(X) we h a v e

+ g]-IXI, where e is a positive number so small that h N ( a r g X ) + j*. Since the set {Rj} has zero linear density (see Len~na i), the function g(X) can be extended from the set C\ UjRj over the whole P. Now consider the functions Jp(X). It follows from their definiX k ~ Sj, 0 g v g m k - I, j = p, and with the aid of the theory of ver"i f y that Jp (v) (X k ) = ~p ( v ) (Xk), Xk =~ Sp, 0 ~ v ~ m k - i. There-

g (~.) -

% (~.) = qa O,).q~, (~.),

where qp(~) are some functions holomorphic

in Sp.

~. ~

$1,,

p = t .....

Theorem I is proved.

Take in each disc Sj an arbitrary point si (j = I, 2, ..) and with the use of the functions h k(argX).lll define the matrix A = { jkY (k, j = i, 2 . . . . ), where

aj~. = exp [hi, (arg s./) j sjj J. Note that A is a Kothe matrix, LEMMA 2. The following D' k > 0 Such that

i.e., 0 ~ aj~: ~ a~+ l (i, k = I, 2 .... ).

inequalities

sup [hk (argO.) [Z I ]
0 and

]naj,~+ 1-c-D~: (j = 1, 2 . . . .

);

(7)

In a:~ ~< inf

[h~+1 (arg ~) ] ~ !] q- DI,.

(] = t, 2, . . .).

(8)

Proof. Prove the first inequality. Let sj* be a point in the closure of Sj at which the fu-~ction h k (argX).IX I attains its maximum on the set Sj. Condition (3) implies the relation limj_~Isj* - sjl/Isj] -- 0. Therefore limj_se [h~ (arg s*) - - h~ (arg sj)] = 0.

(9)

0

Since G k c Gk+1, for some positive zk the relation h k (argA) + z k < hk+ z (argX) (k = l, 2, ... ) holds. From this inequality and relation (9) we obtain hk (arg s3"*) ]s.*l < hk+l (argsj)]sj], j > J0. Therefore, one can find a number D k > 0 for which all j (7) will hold. Inequality (8) can be established in an analogous way. Lemma 2 is proved. Let us introduce several spaces necessary in what follows Let g = (Ej~ sequence of finite-dimensional Banach spaces. For 1