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 Izv. RAN. Ser. Mat., 2018, Volume 82, Issue 1, Pages 225–258 (Mi izv8536)

Sobolev-orthogonal systems of functions associated with an orthogonal system

I. I. Sharapudinovab

a Daghestan Scientific Centre of the Russian Academy of Sciences, Makhachkala

Abstract: For every system of functions $\{\varphi_k(x)\}$ which is orthonormal on $(a,b)$ with weight $\rho(x)$ and every positive integer $r$ we construct a new associated system of functions $\{\varphi_{r,k}(x)\}_{k=0}^\infty$ which is orthonormal with respect to a Sobolev-type inner product of the form
$$\langle f,g \rangle=\sum_{\nu=0}^{r-1}f^{(\nu)}(a)g^{(\nu)}(a)+ \int_{a}^{b} f^{(r)}(t)g^{(r)}(t)\rho(t) dt.$$
We study the convergence of Fourier series in the systems $\{\varphi_{r,k}(x)\}_{k=0}^\infty$. In the important particular cases of such systems generated by the Haar functions and the Chebyshev polynomials $T_n(x)=\cos(n\arccos x)$, we obtain explicit representations for the $\varphi_{r,k}(x)$ that can be used to study their asymptotic properties as $k\to\infty$ and the approximation properties of Fourier sums in the system $\{\varphi_{r,k}(x)\}_{k=0}^\infty$. Special attention is paid to the study of approximation properties of Fourier series in systems of type $\{\varphi_{r,k}(x)\}_{k=0}^\infty$ generated by Haar functions and Chebyshev polynomials.

Keywords: Sobolev-orthogonal systems of functions associated with Haar functions; Sobolev-orthogonal systems of functions associated with Chebyshev polynomials; convergence of Fourier series of Sobolev-orthogonal functions; approximation properties of partial sums of Fourier series of Sobolev-orthogonal functions; convergence of Fourier series of Sobolev-orthogonal polynomials associated with Chebyshev polynomials.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00486-a This paper was written with the support of the Russian Foundation for Basic Research (grant no. 16-01-00486-a).

DOI: https://doi.org/10.4213/im8536

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English version:
Izvestiya: Mathematics, 2018, 82:1, 212–244

Bibliographic databases:

UDC: 517.538
MSC: 41A58, 42C10, 33C47
Revised: 28.07.2016

Citation: I. I. Sharapudinov, “Sobolev-orthogonal systems of functions associated with an orthogonal system”, Izv. RAN. Ser. Mat., 82:1 (2018), 225–258; Izv. Math., 82:1 (2018), 212–244

Citation in format AMSBIB
\Bibitem{Sha18} \by I.~I.~Sharapudinov \paper Sobolev-orthogonal systems of functions associated with an orthogonal system \jour Izv. RAN. Ser. Mat. \yr 2018 \vol 82 \issue 1 \pages 225--258 \mathnet{http://mi.mathnet.ru/izv8536} \crossref{https://doi.org/10.4213/im8536} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2018IzMat..82..212S} \elib{http://elibrary.ru/item.asp?id=32428084} \transl \jour Izv. Math. \yr 2018 \vol 82 \issue 1 \pages 212--244 \crossref{https://doi.org/10.1070/IM8536} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000427245900008} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85043697571} 

• http://mi.mathnet.ru/eng/izv8536
• https://doi.org/10.4213/im8536
• http://mi.mathnet.ru/eng/izv/v82/i1/p225

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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. Sharapudinov I.I., “Sobolev Orthogonal Polynomials Associated With Chebyshev Polynomials of the First Kind and the Cauchy Problem For Ordinary Differential Equations”, Differ. Equ., 54:12 (2018), 1602–1619
2. I. I. Sharapudinov, “Sobolev-orthogonal systems of functions and the Cauchy problem for ODEs”, Izv. Math., 83:2 (2019), 391–412
3. M. G. Magomed-Kasumov, “Sistema funktsii, ortogonalnaya v smysle Soboleva i porozhdennaya sistemoi Uolsha”, Matem. zametki, 105:4 (2019), 545–552
4. R. M. Gadzhimirzaev, “Sobolev-orthonormal system of functions generated by the system of Laguerre functions”, Probl. anal. Issues Anal., 8(26):1 (2019), 32–46
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