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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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