The convolution product is for exponential kernels transforms. Composition is suggested for other transforms. Issue 1 (2nd January 2021)

• Record Type: Journal Article
• Authors: Jerri, A. J.
• Is Part Of: Integral transforms and special functions. Volume 32:Issue 1(2021)
• Page Start: 14
• Page End: 30
• Publication Date: 2021-01-02
• Subjects: Convolution (-generalized) -- convolution-parallel (CP) composition -- sampling theorem -- inverse transform of two transforms product (ITTTP) -- integral transforms -- error bounds -- various complicated analytical results for ITTTP -- generalized hill functions -- brief historical account of the use of convolution -- (CP-) composition -- boundary value problems -- signal analysis; 94A20 -- 44A35; Integral transforms -- Periodicals; Transcendental functions -- Periodicals; Transformations (Mathematics) -- Periodicals; Calculus, Integral -- Periodicals; 515
• DOI: 10.1080/10652469.2020.1786083
• Languages: English
• ISSNs: 1065-2469

Abstract

Abstract : The convolution product is associated with exponential kernel transforms such as the Fourier and Laplace transforms. It's feasibility is due the important property of the simple addition operation for the arguments of the exponential kernels product. This is behind the simple form of the convolution product. No procedure that is remotely close to this exists for the transforms with non-exponential kernels such as that of the Bessel (Hankel) transform. In 1972, we used 'generalized convolution'. Now, we suggest using a 'Convolution Parallel (CP) Composition' instead. This is supported through examples with a variety of integral transforms as well as testimonials from experts in the field including Ruell Churchill and Ian Sneddon. Computations with the CP - Composition are analytically complicated. Thus, there is a need for returning to the numerical integration according to the definition of the Inverse Transform of Two Transforms Product (ITTTP). In 1964, D. Haimo citing Hirschman used a different symbol, associated with the Hankel transform, than the minus sign of the convolution, without giving it a new name. Recently, other authors used the author's original 'generalized translation and convolution.'