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Mat. Zametki, 2008, Volume 83, Issue 5, Pages 722–740 (Mi mz4046)  

This article is cited in 5 scientific papers (total in 5 papers)

Convergence of Biorthogonal Series in the System of Contractions and Translations of Functions in the Spaces $L^p[0,1]$

P. A. Terekhin

Saratov State University named after N. G. Chernyshevsky

Abstract: We obtain conditions for the convergence in the spaces $L^p[0,1]$, $1\le p<\infty$, of biorthogonal series of the form
$$ f=\sum_{n=0}^\infty(f,\psi_n)\varphi_n $$
in the system $\{\varphi_n\}_{n\ge 0}$ of contractions and translations of a function $\varphi$. The proposed conditions are stated with regard to the fact that the functions belong to the space $\mathfrak L^p$ of absolutely bundle-convergent Fourier–Haar series with norm
$$ \|f\|_p^\ast=|(f,\chi_0)| +\sum_{k=0}^\infty 2^{k(1/2-1/p)} (\mspace{2mu}\sum_{n=2^k}^{2^{k+1}-1} |(f,\chi_n)|^p)^{1/p}, $$
where $(f,\chi_n)$, $n=0,1,…$, are the Fourier coefficients of a function $f\in L^p[0,1]$ in the Haar system $\{\chi_n\}_{n\ge 0}$. In particular, we present conditions for the system $\{\varphi_n\}_{n\ge 0}$ of contractions and translations of a function $\varphi$ to be a basis for the spaces $L^p[0,1]$ and $\mathfrak L^p$.

Keywords: biorthogonal series, system of contractions and translations of a function, the space $L^p[0,1]$, bundle convergence of Fourier–Haar series, Haar function, wavelet theory

DOI: https://doi.org/10.4213/mzm4046

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English version:
Mathematical Notes, 2008, 83:5, 657–674

Bibliographic databases:

UDC: 517.51
Received: 19.04.2007
Revised: 11.11.2007

Citation: P. A. Terekhin, “Convergence of Biorthogonal Series in the System of Contractions and Translations of Functions in the Spaces $L^p[0,1]$”, Mat. Zametki, 83:5 (2008), 722–740; Math. Notes, 83:5 (2008), 657–674

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. P. A. Terekhin, “Best approximation of functions in $L_p$ by polynomials on affine system”, Sb. Math., 202:2 (2011), 279–306  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. Sarsenbi A.M. Terekhin P.A., “Riesz Basicity For General Systems of Functions”, J. Funct. space, 2014, 860279  crossref  mathscinet  zmath  isi  scopus
    3. P. A. Terekhin, “Affine Riesz bases and the dual function”, Sb. Math., 207:9 (2016), 1287–1318  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Kh. Kh. Kh. Al-Dzhourani, V. A. Mironov, P. A. Terekhin, “Affinnye sistemy funktsii tipa Uolsha. Polnota i minimalnost”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 16:3 (2016), 247–256  mathnet  crossref  mathscinet  elib
    5. Mironov V.A. Sarsenbi A.M. Terekhin P.A., “Affine Bessel Sequences and Nikishin'S Example”, Filomat, 31:4 (2017), 963–966  crossref  mathscinet  isi  scopus
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