Approximation and Spanning in the Hardy Space,by Affine Systems期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Approximation and Spanning in the Hardy Space,by Affine Systems

Authors:

Institution:

(1) Department of Mathematics, University of Canterbury, Christchurch 8020, New Zealand;(2) Department of Mathematics, University of Illinois, Urbana, IL 61801, USA

Abstract:

We show that every function in the Hardy space can be approximated by linear combinations of translates and dilates of a synthesizer $\psi \in L^1({\bf R}^d)$ , provided only that $\widehat{\psi}(0)=1$ and $\psi$ satisfies a mild regularity condition. Explicitly, we prove scale averaged approximation for each $f \in H^1({\bf R}^d)$ ,

$f(x) = \lim_{J \to \infty} \frac{1}{J} \sum_{j=1}^J \sum_{k \in {\bf Z}^d} c_{j,k} \psi(a_j x - k),$

where

is an arbitrary lacunary sequence (such as $a_j=2^j$

) and the coefficients $c_{j,k}$ are local averages of f. This formula holds in particular if the synthesizer $\psi$ is in the Schwartz class, or if it has compact support and belongs to $L^p$

for some

$1
. A corollary is a new affine decomposition of
<img src=

in terms of differences of $\psi$ .

Keywords:

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