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Perturbation of wavelet frames and Riesz bases. I. (English) Zbl 1101.42308
Summary: Suppose that \(\psi\in L^2(\mathbb{R})\) generates a wavelet frame (resp. Riesz basis) with bounds \(A\) and \(B\). If \(\phi\in L^2(\mathbb{R})\) satisfies \(|\widehat{\psi}(\xi)- \widehat{\phi}(\xi)| < \lambda \frac{|\xi|^\alpha } { ( 1 + | \xi | )^\gamma} \) for some positive constants \(\alpha , \gamma , \lambda\) such that \(1< 1+\alpha < \gamma \) and \(\lambda^2 M< A \), then \(\phi\) also generates a wavelet frame (resp. Riesz basis) with bounds \(A \left ( 1- \lambda \sqrt { M/A} \right )^2 \) and \(B \left ( 1+ \lambda \sqrt {M/B} \right )^2,\) where \(M\) is a constant depending only on \(\alpha,\gamma,\) the dilation step \(a\), and the translation step \(b\).
MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A30 Approximation by other special function classes
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