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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.
UDC:
517.518+
517.982
MSC: Primary
42C15; Secondary
41A65,
42C30,
42C40,
46B15,
46E35 Received: 16.04.2008 and 18.02.2009
DOI:
10.4213/sm5655
Fulltext:
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