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L. Brown; B. Schreiber; B. A. TaylorSpectral synthesis and the Pompeiu problemAnnales de l'institut Fourier, 23 no. 3 ( 1973), p. 125-154, doi: 10.5802/aif.474
Article PDF | Reviews MR 50 #4979 | Zbl 0265.46044 | 5 citations in Cedram
It is shown that every closed rotation and translation invariant subspace $V$ of $C({\bf R}^n)$ or $\delta ({\bf R}^n)$, $n\ge 2$, is of spectral synthesis, i.e. $V$ is spanned by the polynomial-exponential functions it contains. It is a classical problem to find those measures $\mu $ of compact support on ${\bf R}^2$ with the following property: (P) The only function $f\in C({\bf R}^2)$ satisfying $\int _{{\bf R}^2}f\circ \sigma d\mu =0$ for all rigid motions $\sigma $ of ${\bf R}^2$ is the zero function. As an application of the above result a characterization of such measures is obtained in terms of their Fourier-Laplace transforms. Using this characterization, along with asymptotic estimates of the growth of Fourier-Laplace transforms along certain curves, it is shown that property (P) is satisfied by the area measures on a large class of compact regions in the plane. The spectral synthesis theorem also implies Delsarte’s two circle theorem for harmonic functions and other results related to Morera’s converse of the Cauchy integral theorem.
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