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Sectorial Paley-Wiener Theorem

Armen Vagharshakyan

Abstract

By correcting, simplifying and extending a result of Morimoto, we prove a Paley-Wiener type theorem for functions of exponential type in a sector. It serves as a sectorial analogue of Polya's theorem on the indicator of entire functions and improves a result of Dzhrbashyan and Avetisyan by finding the maximal convex set of analytic continuation inside a sector.

Sectorial Paley-Wiener Theorem

Abstract

By correcting, simplifying and extending a result of Morimoto, we prove a Paley-Wiener type theorem for functions of exponential type in a sector. It serves as a sectorial analogue of Polya's theorem on the indicator of entire functions and improves a result of Dzhrbashyan and Avetisyan by finding the maximal convex set of analytic continuation inside a sector.
Paper Structure (5 sections, 7 theorems, 54 equations, 2 figures)

This paper contains 5 sections, 7 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Cauchy formula.
  3. Fourier inversion.
  4. Laplace transform.
  5. Domain of analyticity.

Key Result

Theorem 1.1

The set of entire functions $f$ of exponential type at most $h,$ that are square-integrable on the real axis, coincides with the set of functions $f$ that allow the integral representation

Figures (2)

  • Figure 1: The curve $\Gamma.$
  • Figure 2: The curve $\Gamma^{\prime}.$

Theorems & Definitions (20)

  • Theorem 1.1: Paley-Wiener
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10: Polya
  • ...and 10 more