|
This article is cited in 7 scientific papers (total in 7 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
$$
\sum_{j\in\mathbb N}\lvert\det a_j\rvert^{1/2-1/p}
\biggl(\sum_{k\in\mathbb Z^d}|c_{j,k}|^p\biggr)^{1/p}<\infty.
$$
Bibliography: 19 titles.
Keywords:
affine systems, affine synthesis, frames in a Banach space.
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
Linking options:
https://www.mathnet.ru/eng/sm5655https://doi.org/10.1070/SM2009v200n09ABEH004041 https://www.mathnet.ru/eng/sm/v200/i9/p127
|
Statistics & downloads: |
Abstract page: | 637 | Russian version PDF: | 245 | English version PDF: | 26 | References: | 60 | First page: | 16 |
|