In this paper we present the operational properties of two integral transforms of Fourier type, provide the formulation of convolutions, and obtain eight new convolutions for those transforms. Moreover, we consider applications such as the construction of normed ring structures on L1(R), further applications to linear partial differential equations and an integral equation with a mixed Toeplitz-Hankel kernel. Mathematics Subject Classification (2000). Primary 42B10; Secondary 44A20, 44A35, 47G10.

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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(R n ) are constructed, and explicit solutions of integral equations of convolution type are obtained.