Representation of the fourier transform of distributions in K' p,k' k < 0.

Abstract

In this note we give a structure theorem of the distributions in the space K p,k , k < 0, which is a subspace of the space of distributions which grow no faster than e |x| p , p > 1, and use this structure theorem to give a representation of the Fourier transform of the distributions in these spaces. The Fourier transform of members of several spaces of distributions has been studied by several authors. Gonzalez and Negrin [5] studied the Fourier transform over the spaces S k , k ∈ Z, k < 0 of tempered distributions introduced by Horvath [2]. They have shown that the Fourier transform maps each of the spaces S k , k ∈ Z, k < 0 onto itself, and proved a representation theorem for the usual Fourier transform of members of these spaces. Hayek, Gonzalez and Negrin [3] proved an inversion formula for the distributional Fourier transform on the spaces S k , k ∈ Z, k < 0. They applied their results to obtain a representation on S for any distribution of S k as limit of a sequence of ordinary functions. Gonzalez [4], established a structure theorem of the members of the spaces S k and gave a representation of the Fourier transform of these members. Sohn and Pahk [6] introduced the spaces K p,k , k ∈ Z, k < 0, p > 1, of distributions of exponential growth. Among other things they studied the Fourier transform of members of these spaces and gave an inversion formula for the elements of the spaces. In this work, along the lines of Barrose-Neto [1, proof of Theorem 6.2], we establish a structure theorem for the distributions in the spaces K p,k , k ∈ Z, k < 0, p > 1, then we use this structure theorem to get a representation of the Fourier transform of the elements of these spaces.

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