In this paper we show that if is a convolution operator in , and , then the zeros of the Fourier transform of are of bounded order. Then we discuss relations between the topologies of the space of convolution operators on . Finally, we give sufficient conditions for convergence in the space of convolution operators in and in its dual.
We introduce some spaces of generalized functions that are defined as generalized quotients and Boehmians. The spaces provide simple and natural frameworks for extensions of the Fourier transform.
Mathematics Subject Classification: 44A05, 46F12, 28A78We prove that Dirac’s (symmetrical) delta function and the Hausdorff dimension function build up a pair of reciprocal functions. Our reasoning is based on the theorem by Mellin. Applications of the reciprocity relation demonstrate the merit of this approach.
We give sufficient conditions for the support of the Fourier transform of a certain class of weighted integrable distributions to lie in the region and .
We define various operations on the space of ultra Boehmians like multiplication with certain analytic functions which are Fourier transforms of compactly supported distributions, polynomials, and characters , translation, differentiation. We also prove that the Fourier transform on the space of ultra Boehmians has all the operational properties as in the classical theory.