Spectral synthesis and the Pompeiu problem

@article{Brown1973,
abstract = {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.},
author = {Brown, L., Schreiber, B., Taylor, B. A.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3},
pages = {125-154},
publisher = {Association des Annales de l'Institut Fourier},
title = {Spectral synthesis and the Pompeiu problem},
url = {http://eudml.org/doc/74135},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Brown, L.
AU - Schreiber, B.
AU - Taylor, B. A.
TI - Spectral synthesis and the Pompeiu problem
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 3
SP - 125
EP - 154
AB - 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.
LA - eng
UR - http://eudml.org/doc/74135
ER -