Decomposition theorems and approximation

by a ``floating" system of exponentials

Author:
E. S. Belinskii

Journal:
Trans. Amer. Math. Soc. **350** (1998), 43-53

MSC (1991):
Primary 42A61

DOI:
https://doi.org/10.1090/S0002-9947-98-01556-6

MathSciNet review:
1340169

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Abstract: The main problem considered in this paper is the approximation of a trigonometric polynomial by a trigonometric polynomial with a prescribed *number* of harmonics. The method proposed here gives an opportunity to consider approximation in different spaces, among them the space of continuous functions, the space of functions with uniformly convergent Fourier series, and the space of continuous analytic functions. Applications are given to approximation of the Sobolev classes by trigonometric polynomials with prescribed number of harmonics, and to the widths of the Sobolev classes. This work supplements investigations by Maiorov, Makovoz and the author where similar results were given in the integral metric.

**[St]**S. B. Stechkin,*On the best approximation of given classes of functions by arbitrary polynomials*, Uspekhi Matematicheskikh Nauk**9**(1) (1954), 133-134 (Russian).**[Is]**R. S. Ismagilov,*Widths of set in normed linear spaces and approximation of functions by trigonometric polynomials*, Uspecki Matematicheskikh Nauk**29**(3) (1974), 161-178 (Russian), English translation in Russian Math. Surveys**28**(3) (1974). MR**53:11284****[Mr1]**V. E. Maiorov,*On linear widths of Sobolev classes and chains of extremal subspaces*, Matematicheski[??]i Sbornik**113**(1980), 437-463;**119**(1982), 301; English translations in Math. USSR Sb.**41**(1982);**47**(1984). MR**82j:41022**; MR**84b:41021****[Mr2]**V. E. Maiorov,*Trigonometric widths of Sobolev classes in the space*, Matematicheskie Zametki**40**(2) (1986), 161-173; English translation in Math. Notes**40**(1986). MR**87k:46072****[Mr3]**V. E. Maiorov,*On the best approximation of classes in the space*, Matematicheskie Zametki**19**(1976), 699-706; English translation in Math. Notes**19**(1976). MR**54:10946****[Mk]**Y. Makovoz,*On trigonometric -widths and their generalizations*, J. Approx. Theory**41**(1984), 361-366. MR**86g:41038****[Be1]**E. S. Belinskii,*Approximation of periodic functions by a ``floating" system of exponentials*, Studies in the Theory of Functions of Several Real Variables (Y. A. Brudnyi, ed.), Yaroslav. Gos. Univ., Yaroslavl, 1984, pp. 10-24 (Russian). MR**88j:42002****[Be2]**E. S. Belinskii,*Approximation by a ``floating" system of exponentials on classes of smooth periodic functions*, Matematischeski[??]i Sbornik**132**(1987), 20-27; English translation in Math. USSR Sb.**60**(1988). MR**88d:42001****[Z]**A. Zygmund,*Trigonometric series*, 2nd ed., Cambridge Univ. Press, Cambridge, 1959. MR**21:6498****[Bo]**Y. Bourgain,*Bounded orthogomal systems and the -set problem*, Acta Math.**162**(3-4) (1989), 227-245. MR**90h:43008****[G1]**E. D. Gluskin,*Extremal properties of orthogonal parallelepipeds and their application to the geometry of Banach spaces*, Matematischeski[??]iSbornik**136**(1988), 85-96; English translation in Math. USSR Sb.**64**(1989). MR**89j:46106****[Sp]**J. Spencer,*Six standard deviations suffice*, Trans. Amer. Math. Soc.**289**(2) (1985), 679-706.MR**86k:05005****[Ki1]**S. V. Kislyakov,*Quantitative aspect of the ``corrigible" theorems*, Investigations on Linear Operators and Function Theory, Zapiski LOMI**92**(1979), 182-191. (Russian) MR**82c:28012****[Ki2]**S. V. Kislyakov,*Fourier coefficints of boundary values of functions that are analytic in the disk and bidisk*, Spectral Theory and Functional Operators II, Trudy Math. Inst. Steklov**155**(1981) 77-94; English translation in Proc. Steklov Inst. Math. 1983, no. 1 (155). MR**83a:42005****[Ho]**K. Höllig,*Approximationszahlen von Sobolev-Einbettungen*, Mathematische Annalen**242**(1979), 273-281. MR**80j:46051****[Kr]**M. A. Krasnoselskii and Y. B. Rutitskii,*Convex functions and Orlicz spaces*, Fizmatgiz, Moscow, 1958; English translation, Noordhoff, Groningen, 1961. MR**21:5144**; MR**23:A4016**

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Additional Information

**E. S. Belinskii**

Affiliation:
Department of Mathematics, Technion, 32000, Haifa, Israel

Address at time of publication:
Department of Mathematics, University of Zimbabwe, P. O. Box MP167, Mount Pleasant, Harare, Zimbabwe

Email:
belinsky@maths.uz.zw

DOI:
https://doi.org/10.1090/S0002-9947-98-01556-6

Keywords:
Approximation,
width

Received by editor(s):
March 13, 1995

Additional Notes:
This research was supported by the Israeli Ministry of Science and the Arts through the Ma’agara program for absorption of immigrant mathematicians at the Technion, Israel Institute of Technology

Article copyright:
© Copyright 1998
American Mathematical Society