Convolutions for the Fourier transforms with geometric variables and applications

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 L1(Rn) are constructed, and explicit solutions of integral equations of convolution type are obtained (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)