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Keywords:
Fourier transform; Lebesgue space; tempered distribution; generalised function; Banach space; continuous primitive integral
Fourier transform; Lebesgue space; tempered distribution; generalised function; Banach space; continuous primitive integral
Summary:
For each $f\in L^p({\mathbb R)}$ ($1\leq p<\infty $) it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each $p$, a norm is defined so that the space of Fourier transforms is isometrically isomorphic to $L^p({\mathbb R)}$. There is an exchange theorem and inversion in norm.
References:
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