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A Fourier Inversion Theorem for Normal Functions

Tristram de Piro

Abstract

We prove an inversion theorem for the Fourier transform defined for normal functions, in the case when such functions are of moderate decrease, and in dimensions 2 and 3. This improves on Carleson's general almost everywhere convergence result for square integrable functions to everywhere convergence, in the special case of normal functions of moderate decrease. The class of normal functions appear in Physics and the everywhere convergence is important for further analysis of wave equations and no radiation properties of electromagnetic fields.

A Fourier Inversion Theorem for Normal Functions

Abstract

We prove an inversion theorem for the Fourier transform defined for normal functions, in the case when such functions are of moderate decrease, and in dimensions 2 and 3. This improves on Carleson's general almost everywhere convergence result for square integrable functions to everywhere convergence, in the special case of normal functions of moderate decrease. The class of normal functions appear in Physics and the everywhere convergence is important for further analysis of wave equations and no radiation properties of electromagnetic fields.
Paper Structure (13 theorems)

This paper contains 13 theorems.

Key Result

Lemma 1

Let $f:\mathcal{R}^{2}\rightarrow\mathcal{R}$ be smooth and quasi split normal, then $\mathcal{F}(f)\in L^{1}(\mathcal{R}^{2})$ and is of rapid decay, in the sense that, for $|\overline{k}|>1$, $k_{1}\neq 0,k_{2}\neq 0$$|\mathcal{F}(f)(\overline{k})|\leq {C_{n}\over |\overline{k}|^{n}}$ where $C_{n}

Theorems & Definitions (29)

  • Lemma 1
  • proof
  • Definition 2
  • Definition 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 19 more