Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [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 $W_{p}^{r}$ in the space $L_{q}$, 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 $W_{1}^{r}(I^{s})$ in the space $L_{\infty }(I^{s})$, Matematicheskie Zametki 19 (1976), 699-706; English translation in Math. Notes 19 (1976). MR 54:10946
  • [Mk] Y. Makovoz, On trigonometric $n$-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 $\Lambda (p)$-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

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 42A61

Retrieve articles in all journals with MSC (1991): 42A61


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

American Mathematical Society