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Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 5, Pages 115–144 (Mi izv3562)  

This article is cited in 1 scientific paper (total in 1 paper)

Linear algorithms of affine synthesis in the Lebesgue space $L^1[0,1]$

P. A. Terekhin

Saratov State University named after N. G. Chernyshevsky

Abstract: We prove that there are no linear algorithms of affine synthesis for affine systems in the Lebesgue space $L^1[0,1]$ with respect to the model space $\ell^1$, although the corresponding affine synthesis problem has a positive solution under the most general assumptions. At the same time, by imposing additional conditions on the generating function of the affine system, we can give an explicit linear algorithm of affine synthesis in the Lebesgue space when the model space is that of the coefficients of the system. This linear algorithm generalizes the Fourier–Haar expansion into orthogonal series.

Keywords: affine system, affine synthesis, frames in Banach spaces, representation system, Fourier–Haar series, primary space.

DOI: https://doi.org/10.4213/im3562

Full text: PDF file (657 kB)
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English version:
Izvestiya: Mathematics, 2010, 74:5, 993–1022

Bibliographic databases:

UDC: 517.51+517.98
MSC: 41A15, 42C15, 42C40, 46B15, 47B37
Received: 05.08.2008

Citation: P. A. Terekhin, “Linear algorithms of affine synthesis in the Lebesgue space $L^1[0,1]$”, Izv. RAN. Ser. Mat., 74:5 (2010), 115–144; Izv. Math., 74:5 (2010), 993–1022

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    This publication is cited in the following articles:
    1. 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
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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