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A spectral Paley-Wiener theorem

Monatshefte für Mathematik volume 116, pages 1–11 (1993)Cite this article

Abstract

The Fourier inversion formula in polar form is\(f(x) = \int_0^\infty {P_\lambda } f(x)d\lambda \) for suitable functionsf on ℝn, whereP λ f(x) is given by convolution off with a multiple of the usual spherical function associated with the Euclidean motion group. In this form, Fourier inversion is essentially a statement of the spectral theorem for the Laplacian and the key question is: how are the properties off andP λ f related? This paper provides a Paley-Wiener theorem within this avenue of thought generalizing a result due to Strichartz and provides a spectral reformulation of a Paley-Wiener theorem for the Fourier transform due to Helgason. As an application we prove support theorems for certain functions of the Laplacian.

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Authors and Affiliations

  1. Department of mathematics, University of Maine, Neville Hall, 04469-0122, Orono, Maine, USA

    William O. Bray

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  1. William O. Bray
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Bray, W.O. A spectral Paley-Wiener theorem. Monatshefte für Mathematik 116, 1–11 (1993). https://doi.org/10.1007/BF01388416

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  • DOI: https://doi.org/10.1007/BF01388416

Keywords

  • Fourier
  • Fourier Transform
  • Convolution
  • Spherical Function
  • Inversion Formula