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.

摘要

在“特殊仿射傅立叶变换的卷积和乘积定理”一文中 [In: Nashed MZ, Li X, editors. 正交多项式和 q 系列的前沿。世界科学；2018 年。119-137]，在特殊仿射傅立叶变换领域提出了卷积结构。在继续研究的过程中，我们引入了一种新颖的积分变换，即通过特殊仿射卷积结合了著名的特殊仿射傅立叶变换和小波变换的优点，创造了特殊仿射小波变换。初步分析包括基本性质的推导、莫亚尔原理、反演公式和极差定理。随后，我们获得了海森堡不确定性原理的温和扩展，并为特殊仿射傅立叶变换开发了皮特不等式的类似物。此外，我们推导出特殊仿射小波变换的海森堡型不确定原理。最后，我们通过引入特殊仿射小波变换组合的概念来扩展本研究的范围。