Convolutions for the Fourier transforms with geometric variables and applications

Abstract

Abstract This paper gives a general formulation of convolutions for arbitrary linear operators from a linear space to a commutative algebra, constructs three convolutions for the Fourier transforms with geometric variables and four generalized convolutions for the Fourier‐cosine, Fourier‐sine transforms. With respect to applications, by using the constructed convolutions normed rings on L 1 ( R n ) are constructed, and explicit solutions of integral equations of convolution type are obtained (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)