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We first construct a space (cn) whose elements are test functions defined in cn = n {∞}, the one point compactification of n, that have a thick expansion at infinity of special logarithmic type, and its dual space ′(cn), the space of sl-thick distributions. We show that there is a canonical projection of ′(cn) onto ′(n). We study several sl-thick distributions and consider operations in ′(cn). We define and study the Fourier transform of thick test functions of (n) and thick tempered distributions of ′(n). We construct isomorphisms : ′(n) →′(cn), : ′(cn) → ′(n), that extend the Fourier transform of tempered distributions, namely, = and =, where are the canonical projections of ′(n) or ′(cn) onto ′(n). We determine the Fourier transform of several finite part regularizations and of general thick delta functions.

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Analysis and Applications

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