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Convolutions for the Fourier transforms with geometric variables and applications. (English) Zbl 1226.43003
The authors consider generalised convolutions in the sense of Kakichev associated with the Euclidean Fourier transform and some of its variants: the Fourier transform composed with a shift, with dilations, with inversion, and the Fourier-cosine and Fourier-sine transforms. The authors show that \(L^1(\mathbb R^n)\) equipped with these generalised convolutions is a normed ring and determine when it is commutative. As application, necessary and sufficient conditions for solving integral equations of convolution types, as well as the solutions, are given.

43A32 Other transforms and operators of Fourier type
44A35 Convolution as an integral transform
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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