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Fourier Transforms

In: Introduction to Hyperfunctions and Their Integral Transforms

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  • Urs Graf

Abstract

From the outset we shall explore the relation between Laplace and Fourier transforms. The subclass of hyperfunctions of slow growth S( ℝ) and their Fourier transforms are introduced. It will be shown that the Fourier transform of a hyperfunction can be computed by evaluating the Laplace transform of two rightsided hyperfunctions. This fact is exploited to the hilt in the sequel, almost all Fourier transforms in this book are computed via Laplace transforms. The inverse Fourier transformation and the important Reciprocity Rule are formulated for hyperfunctions. All operational rules for the Fourier transform of hyperfunctions are carefully established. Conditions for the validity of the convolution property of Fourier transformation are stated. Many concrete examples of Fourier transforms of hyperfunctions are presented. The chapter terminates with Poisson’s summation formula and some applications to differential and integral equations.

Suggested Citation

  • Urs Graf, 2010. "Fourier Transforms," Springer Books, in: Introduction to Hyperfunctions and Their Integral Transforms, chapter 0, pages 241-274, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0346-0408-6_4
    DOI: 10.1007/978-3-0346-0408-6_4
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