Wavelet transforms on Gelfand-Shilov spaces and concrete examples (Q2012332)

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.