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Wavelet transforms on Gelfand-Shilov spaces and concrete examples. (English) Zbl 1407.42018

The Gelfand-Shilov spaces \(S_{\nu}^{\mu}\) consist of, roughly speaking, functions with exponential decay at infinity of order of \(e^{-|x|^{1/\nu}}\) and decay of order \(e^{-|\xi|^{1/\mu}}\) for its Fourier transform. The authors defined tighter Gelfand-Shilov-type spaces \(S_{\nu}^{\mu,\delta }\) by further conditions of the vanishing moments, which requires that the Fourier transform of a function vanishes at \(0\) like \(e^{-1/|\xi|^{1/\delta }}\). For both a signal and a mother wavelet in \(S_{\nu}^{\mu,\delta }\) the paper presents some estimates for the continuous wavelet transform, which imply its continuity in the Gelfand-Shilov-type spaces. Some concrete examples of the Fourier transforms and the wavelet transforms in the Gelfand-Shilov-type spaces are computed.

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
46F05 Topological linear spaces of test functions, distributions and ultradistributions
65T60 Numerical methods for wavelets
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