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On the stability of wavelet frames. (Chinese. English summary) Zbl 0933.42026

Summary: The theory of frames is very important for wavelet analysis. For \(\varphi\in L^2(\mathbb{R})\) and \(a>1\), \(b>0\), I. Daubechies gave a sufficient condition ensuring \(\{a^{j/2}\varphi(a^jx- kb):j,k\in \mathbb{Z}\}\) to be a frame for \(L^2(\mathbb{R})\). Recently, much effort has spent on the study of the stability of wavelet frames. In this paper, after obtaining a multivariate version of Kadec’s \(1/4\)-theorem, we study the stability of wavelet frames when \(\varphi\), \(\{a^j\}\) and \(\{k\}\) have some perturbation simultaneously. In particular, we study the effect of the perturbation to \(\{a^j\}\).

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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