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Approximation and Spanning in the Hardy Space, by Affine Systems

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Constructive Approximation Aims and scope

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 \(a_j\) 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 \(1<p<\infty\). A corollary is a new affine decomposition of \(H^1\) in terms of differences of \(\psi\).

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Authors and Affiliations

  1. Department of Mathematics, University of Canterbury, Christchurch 8020, New Zealand

    H.-Q. Bui

  2. Department of Mathematics, University of Illinois, Urbana, IL 61801, USA

    R.S. Laugesen

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  1. H.-Q. Bui
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  2. R.S. Laugesen
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Correspondence to H.-Q. Bui or R.S. Laugesen.

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Bui, HQ., Laugesen, R. Approximation and Spanning in the Hardy Space, by Affine Systems. Constr Approx 28, 149–172 (2008). https://doi.org/10.1007/s00365-006-0672-1

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  • DOI: https://doi.org/10.1007/s00365-006-0672-1

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