RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Forthcoming papers Archive Impact factor Guidelines for authors License agreement Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Trudy MIAN: Year: Volume: Issue: Page: Find

 Tr. Mat. Inst. Steklova, 2010, Volume 269, Pages 8–30 (Mi tm2904)

Wavelet approximation and Fourier widths of classes of periodic functions of several variables. I

D. B. Bazarkhanov

Institute of Mathematics, Almaty, Kazakhstan

Abstract: We obtain characterizations (and prove the corresponding equivalence of norms) of function spaces $\mathbf B^{sm}_{pq}(\mathbb I^k)$ and $\mathbf L^{sm}_{pq}(\mathbb I^k)$ of Nikol'skii–Besov and Lizorkin–Triebel types, respectively, in terms of representations of functions in these spaces by Fourier series with respect to a multiple system $\mathcal W^\mathbb I_m$ of Meyer wavelets and in terms of sequences of the Fourier coefficients with respect to this system. We establish order-sharp estimates for the approximation of functions in $B^{sm}_{pq}(\mathbb I^k)$ and $L^{sm}_{pq}(\mathbb I^k)$ by special partial sums of these series in the metric of $L_r(\mathbb I^k)$ for a number of relations between the parameters $s,p,q,r$, and $m$ ($s=(s_1,…,s_n)\in\mathbb R^n_+$, $1\leq p,q,r\leq\infty$, $m=(m_1,…,m_n)\in\mathbb N^n$, $k=m_1+…+m_n$, and $\mathbb I= \mathbb R$ or $\mathbb T$). In the periodic case, we study the Fourier widths of these function classes.

Full text: PDF file (341 kB)
References: PDF file   HTML file

English version:
Proceedings of the Steklov Institute of Mathematics, 2010, 269, 2–24

Bibliographic databases:

Document Type: Article
UDC: 517.518.224+517.518.837
Received in January 2010

Citation: D. B. Bazarkhanov, “Wavelet approximation and Fourier widths of classes of periodic functions of several variables. I”, Function theory and differential equations, Collected papers. Dedicated to Academician Sergei Mikhailovich Nikol'skii on the occasion of his 105th birthday, Tr. Mat. Inst. Steklova, 269, MAIK Nauka/Interperiodica, Moscow, 2010, 8–30; Proc. Steklov Inst. Math., 269 (2010), 2–24

Citation in format AMSBIB
\Bibitem{Baz10} \by D.~B.~Bazarkhanov \paper Wavelet approximation and Fourier widths of classes of periodic functions of several variables.~I \inbook Function theory and differential equations \bookinfo Collected papers. Dedicated to Academician Sergei Mikhailovich Nikol'skii on the occasion of his 105th birthday \serial Tr. Mat. Inst. Steklova \yr 2010 \vol 269 \pages 8--30 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm2904} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2729970} \zmath{https://zbmath.org/?q=an:1219.42025} \elib{http://elibrary.ru/item.asp?id=15109748} \transl \jour Proc. Steklov Inst. Math. \yr 2010 \vol 269 \pages 2--24 \crossref{https://doi.org/10.1134/S0081543810020021} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000281705900002} \elib{http://elibrary.ru/item.asp?id=15367530} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77956636115} 

• http://mi.mathnet.ru/eng/tm2904
• http://mi.mathnet.ru/eng/tm/v269/p8

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D. B. Bazarkhanov, “Estimates for the widths of classes of periodic functions of several variables – I”, Eurasian Math. J., 1:3 (2010), 11–26
2. Hansen M. Sickel W., “Best M-Term Approximation and Sobolev-Besov Spaces of Dominating Mixed Smoothness-the Case of Compact Embeddings”, Constr. Approx., 36:1 (2012), 1–51
3. Bazarkhanov D.B., “Wavelet Approximation and Fourier Widths of Classes of Periodic Functions of Several Variables. II”, Anal. Math., 38:4 (2012), 249–289
4. D. B. Bazarkhanov, “Nonlinear approximations of classes of periodic functions of many variables”, Proc. Steklov Inst. Math., 284 (2014), 2–31
5. Sh. A. Balgimbaeva, T. I. Smirnov, “Otsenki poperechnikov Fure klassov periodicheskikh funktsii so smeshannym modulem gladkosti”, Tr. IMM UrO RAN, 21, no. 4, 2015, 78–94
6. D. B. Bazarkhanov, “Nonlinear trigonometric approximations of multivariate function classes”, Proc. Steklov Inst. Math., 293 (2016), 2–36
7. Balgimbayeva Sh.A., “Hyperbolic Cross Approximation With Respect to Wavelet System With Compact Supports”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA2017), AIP Conference Proceedings, 1880, eds. Kalmenov T., Sadybekov M., Amer Inst Physics, 2017, UNSP 030005
8. D. B. Bazarkhanov, “Lineinoe vosstanovlenie psevdodifferentsialnykh operatorov na klassakh gladkikh funktsii na m-mernom tore. I”, Tr. IMM UrO RAN, 24, no. 4, 2018, 57–79
•  Number of views: This page: 330 Full text: 9 References: 88