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A spectral Paley-Wiener theorem. (English) Zbl 0803.43001

The Fourier inversion formula in polar coordinates is of the form \(f(x)= \int_ 0^ \infty P_ \lambda f(x) d\lambda\) where \(P_ \lambda f(x)= (2\pi)^{-n} \lambda^{n-1} \int_{S^{n-1}} \widetilde{f} (\lambda\omega) e^{i(\lambda\omega)} d\omega\) and \(\widetilde{f}\) the Fourier transform of \(f\). In [J. Funct. Anal. 87, 51-148 (1989; Zbl 0694.43008)] R. Strichartz investigates the question how do properties of \(f\) relate to ones of \(P_ \lambda f\), and gives a Paley- Wiener theorem in odd \(n\). Roughly speaking, \(P_ \lambda f(x)\) for \(f\in C_ c^ \infty (\mathbb{R}^ n)\) is characterized as an entire eigenfunction \(f_ \lambda(x)\) of the Laplacian \(\Delta\) with eigenvalue \(-\lambda^ 2\) and \(| f_ \lambda (x)| \leq C_ N |\lambda |^{n-1} (1+| \lambda |)^{-N} e^{(R+| x|)| {\mathfrak I}x|}\) for each \(N\), and the failure in even \(n\) is connected with the validity of Huygen’s principle. In this paper the author generalizes this result to all \(n\) by replacing \(C_ N\) with a continuous function \(C_ N(| x|)\), and provides a spectral reformulation of a Paley-Wiener theorem for \(\widetilde{f}\) due to Helgason. As an application support theorems for certain functions of \(\Delta\) are obtained.

MSC:

43A85 Harmonic analysis on homogeneous spaces
22E30 Analysis on real and complex Lie groups
58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Citations:

Zbl 0694.43008
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References:

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