Spectral synthesis in sobolev spaces

CHAPTER 6 APPROXIMATION Certainly, almost all the problems collected in our problem book can be interpreted as approximation problems. Therefore, the selection of the problems included in this chapter has been possibly even more conditional than in the other chapters. However, the not too voluminous material of this chapter is already sufficient to illustrate some important directions in the contemporary theory of approximations. The first four sections (1.6-4.6) are closely related to the ideas of the previous chapters, and this is hardly the result of a coincidence of circumstances: Responding to the internal needs of the theory of functions, the approximation problems constitute, in addition, the essence of harmonic (and thus, also of spectral) analysis. Obviously, the sufficiently wide scattering of the contents of Secs. 1.6-4.6 will not mislead the careful reader: It is not so important what and with what we try to approximate (by rational functions or by exponentials with prescribed frequencies, by polynomials with weights or by ~-measures inside a certain spectral subspace), but important are the inner meaning, the purpose and the stimulating cause for this -- to separate the simplest "harmonics" with respect to a certain action and then to reestablish from them their generating object. That variant of the spectral synthesis which is considered in Sec. 1.6 is aimed, nevertheless, at approximation problems in integral metrics by analytic functions. Section 2.6, oriented towards problems of the same type, discusses integral estimates of the derivative of the conformal mapping of a plane domain. In the final analysis, the interest in such estimates is determined by the problems of the "weak invertibility" (regarding this, see Chap. 5) and has its roots in the "alveolar effect," described for the first time by M. V. Keldysh. Sections 1.6 and 2.6 are also closely related with the ideas of the theory of potential and with various characteristics of "thinness," "rarity" of sets, in accordance with the tendency which is characteristic for the contemporary study of the approximation properties of analytic functions. At the end of the chapter there is a section in which one presents conjectures regarding certain variants of the Wiener rarefying criteria. Problem 3.6 requires little commentary: To this problem there is devoted an extensive and interesting literature where the reader will find not only a well-developed analytic technique but sometimes also literal answers to the questions of Sec. 3.6; we mention only the following publications, not listed in that section: A. Beurling, Lectures on Quasianalyticity, Princeton (1962), preprint; S. V. Khrushchev, Dokl. Akad. Nauk SSSR, 214, No. 3, 524-527 (1974); Tr. Mat. Inst. im. V. A. Steklova, 130, 124-195 (1978), and also A. L. Vol'berg, Dokl. Akad. Nauk SSSR (1978). For Sec. 4.6 we mention the classical paper of A. Beurling and P. Malliavin, Acta Math., 118, Nos. I-2, 79-93 (1967). Sections 5.6 and 6.6 reflect the recently noticed vivid interest in the classical topic of the "Pad~fractions$' whose direction gives us some hope regarding other problems of our collection (see Sec. 6.4). The circle of ideas of P. L. Chebyshev, the eternal problems of the theory of approximation, is reflected also in the pages of this chapter: Through spectral theory, Hankel operators and the s-numbers of linear operators, it is undoubtedly present in Secs. 7.6 and 8.6. From the other important directions of the theory of approximations, only one is represented here: In Sec. 9.6 problems arising in complex analysis under the effect of the ideas of the theory of Banach algebras are discussed.

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1.6.

SPECTRAL SYNTHESIS IN SOBOLEV SPACES*

Let X be a Banach space of functions Sobolev spaces

W$

(function classes) on I~

We have in mind the

, ~ $ p < ~ .,s~Z+ , or the spaces obtained from Sobolev spaces by interpola-

tion (Bessel potential

spaces L~, s > 0, and Besov spaces np,q u s , s > 0).

Then the dual space

X' is a space of distributions. We say that a closed set K in ~ admits X-spectral synthes~s if every T in X' that has support in K can be approximated arbitrarily closely in X' by linear combinations of measures and derivatives of order <S of measures with support in K. Problem.

Do all closed sets admit X-spectral

synthesis for the above spaces?

The problem can also be given a dual formulation.

If v is a measure with support in K

such that a partial derivative Dkv belongs to X', then one can define X.

Then K admits X-spectral

synthesis if every f such that -I~ = 0

ID ~

for all f in

for all such v and

all such multiindices k can be approximated arbitrarily closely in X by test functions that vanish on some neighborhood of K. The problem is of course analogous to the famous spectral synthesis problem of Beurling but in the case of W~ this terminology was introduced by Fuglede. He also observed that the so-called fine Dirichlet problem in a domain D for an elliptic partial differential equation of order 2s always has a unique solution if and only if the complement of D admits W~-spectral synthesis. See [I; IX, 5.1]. In the case of W~ the problem appeared and was solved in the work of Havin [2] and Bagby [3] in connection with the problem of approximation in L p by analytic functions. For W~ the solution appears already in the work of Beurling and Deny [4]. In fact, in these spaces all closed sets have the spectral synthesis property. This result, which can be extended to L~, 0 < s < I, depends mainly on the fact that these spaces are closed under truncations. When s > I this is no longer true, and the problem is more complicated. Using potential theoretic methods the author [5] has given sufficient conditions for sets to admit spectral synthesis in ), 5 ~ + These conditions are so weak that they are satisfied for all closed sets if p > max (d/2, 2 -- I/d), thus in particular if p = 2 and d = 2 or 3. There are also some still unpublished results for L~ and B~ 'p showing, for example, that sets that satisfy a cone condition have the spectral synthesis property. Otherwise, for general spaces the author is only aware of the work of Triebel [6] where he proved, extending earlier results of Lions and Magenes, that the boundary of a C ~ domain admits spectral synthesis for L~ and B~ 'p Editors' Note.

It seems worth noting that when d = I some other details concerning the

problem of synthesis in W~(~), W $ ~]T) are known (see J.-P. Kahane, S~minaire N. Bourbaki 1966, Nov.; Eo G. Akut0wicz, C. R. Acad. Sci., 256, No. 25, 5268-5270 (1963); Ann. Sci. Ecole Norm. Sup., 82, No. 3, 297-325 (1965); IIi. J. Math., 14 No. 2 198-204 (1970); N. M. Osadchin, Ukr. Mat. Zh., 26, No. 5, 669-670 (1974). LITERATURE CITED 1 9

2. 3.

B.-W. Schulze and G. Wildenhain, Methoden der Potentialtheorie fHr elliptische Differentialgleichungen beliebiger Ordnung, Akademie-Verlag, Berlin (1977). V. P. Khavin, "Approximationin the mean by analytic functions," Dokl. Akad. Nauk SSSR, 178, 1025-1028(1968). T. Bagby, "Quasi topologies and rational approximation," J. Funct. Anal., 10, 259-268 (1972).

*LARS INGE HEDBERG. Stockholm, Sweden.

Department of Mathematics,

University of Stockholm, Box 6701, S-11385

2209

4.

5.

A. Beurling and J. Deny, "Dirichlet spaces," Proc. Nat. Acad. Sci., 4_~5, 208-215 (1959). L. I. Hedberg, "Two approximation problems in function spaces," Ark. Mat., 16, 5]-8]

(1978). 6.

2210

H. Triebel, "Boundary values for Sobolev-spaces with weights. Sc. Norm. Sup. Pisa, 3, No. 27, 73-96 (1973).

Density of D(~)," Ann.