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Mat. Sb., 2009, Volume 200, Number 9, Pages 127–146 (Mi msb5655)  

This article is cited in 4 scientific papers (total in 4 papers)

Banach frames in the affine synthesis problem

P. A. Terekhin

Saratov State University named after N. G. Chernyshevsky

Abstract: We consider the problem of representing functions $f\in L^p(\mathbb R^d)$ by a series in elements of the affine system
$$ \psi_{j,k}(x)=\lvert\det a_j\rvert^{1/2}\psi(a_jx-bk), \qquad j\in\mathbb N, \quad k\in\mathbb Z^d. $$
The corresponding representation theorems are established on the basis of the frame inequalities
$$ A\|g\|_q\le\|\{(g,\psi_{j,k})\}\|_Y\le B\|g\|_q $$
for the Fourier coefficients $\displaystyle(g,\psi_{j,k})=\int_{\mathbb R^d}g(x)\psi_{j,k}(x) dx$ of functions $g\in L^q(\mathbb R^d)$, $1/p+1/q=1$, where ${\|\cdot\|}_Y$ is the norm in some Banach space of number families $\{y_{j,k}\}$ and $0<A\le B<\infty$ are constants.
In particular, it is proved that if the integral of a function $\psi\in L^1\cap L^p(\mathbb R^d)$, $1<p<\infty$, is nonzero, so $\displaystyle\int_{\mathbb R^d}\psi(x) dx\ne0$ and the system of translates $\{\psi(x-bk):k\in\mathbb Z^d\}$ is $p$-Besselian in the space $L^p(\mathbb R^d)$, then for any function $f\in L^p(\mathbb R^d)$ we have the representation
$$ f=\sum_{j\in\mathbb N}\sum_{k\in\mathbb Z^d}c_{j,k}\psi_{j,k}, $$
where the coefficients satisfy the condition
$$ \sum_{j\in\mathbb N}\lvert\det a_j\rvert^{1/2-1/p} (\sum_{k\in\mathbb Z^d}|c_{j,k}|^p)^{1/p}<\infty. $$

Bibliography: 19 titles.

Keywords: affine systems, affine synthesis, frames in a Banach space.

DOI: https://doi.org/10.4213/sm5655

Full text: PDF file (585 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2009, 200:9, 1383–1402

Bibliographic databases:

UDC: 517.518+517.982
MSC: Primary 42C15; Secondary 41A65, 42C30, 42C40, 46B15, 46E35
Received: 16.04.2008 and 18.02.2009

Citation: P. A. Terekhin, “Banach frames in the affine synthesis problem”, Mat. Sb., 200:9 (2009), 127–146; Sb. Math., 200:9 (2009), 1383–1402

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. P. A. Terekhin, “Linear algorithms of affine synthesis in the Lebesgue space $L^1[0,1]$”, Izv. Math., 74:5 (2010), 993–1022  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. P. A. Terekhin, “Frames in Banach Spaces”, Funct. Anal. Appl., 44:3 (2010), 199–208  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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. K. S. Speranskii, P. A. Terekhin, “O suschestvovanii freimov v prostranstve Khardi, postroennykh na osnove yadra Sege”, Izv. vuzov. Matem., 2019, no. 2, 57–68  mathnet  crossref
  • Математический сборник Sbornik: Mathematics (from 1967)
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