Progressive Gelfand-Shilov spaces and wavelet transforms
The authors deal with the so-called progressive Gelfand-Shilov spaces \(S_\mu ^{\nu ,+}(\mathbb{R})\), constituted by all functions \(f\) in the usual Gelfand-Shilov space \(S_\mu ^\nu (\mathbb{R})\) such that the support of their Fourier transform \(\widehat{f}\) is contained in \([0,\infty )\). They deduce characterizations of the elements of \(S_\mu ^{\nu ,+}(\mathbb{R})\) in terms of the behavior of \(f\) and \(\widehat{f}\) from the corresponding ones for \(S_\mu ^\nu\) and give an additional localization property that shows that functions in \(S_\mu ^{\nu ,+}(\mathbb{R})\) are almost exponentially strip localized. Also, they establish continuity properties for the wavelet transform on \(S_\mu ^{\nu ,+}(\mathbb{R})\times S_\mu ^{\nu ,+}(\mathbb{R})\) and analyse the inverse wavelet transform. Finally, slightly faster decay on functions in progressive Gelfand-Shilov spaces is considered. That allows them to reformulate the previous results in the new context and obtain a control of the decay of the wavelet transform independently in each variable.
scientific article; zbMATH DE number 6102746
scientific article