A convolution-based special affine wavelet transform
Firdous A. Shah, Azhar Y. Tantary, Ahmed I. Zayed
Abstract
In the article ‘Convolution and product theorems for the special affine Fourier transform’ [In: Nashed MZ, Li X, editors. Frontiers in orthogonal polynomials and q-series. World Scientific; 2018. p. 119–137], a convolution structure is presented in the realm of the special affine Fourier transform. In continuation of the study, we introduce a novel integral transform coined the special affine wavelet transform by combining the merits of the well-known special affine Fourier and wavelet transforms via the special affine convolution. The preliminary analysis encompasses the derivation of fundamental properties, Moyal's principle, inversion formula and range theorem. Subsequently, we obtain a mild extension of Heisenberg's uncertainty principle and also develop an analogue of Pitt's inequality for the special affine Fourier transform. In addition, we derive a Heisenberg-type uncertainty principle for the special affine wavelet transform. Finally, we extend the scope of the present study by introducing the notion of composition of special affine wavelet transforms.