Spectral synthesis and the Pompeiu problem

L. Brown; B. Schreiber; B. A. Taylor

Annales de l'institut Fourier (1973)

  • Volume: 23, Issue: 3, page 125-154
  • ISSN: 0373-0956

Abstract

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It is shown that every closed rotation and translation invariant subspace V of C ( R n ) or δ ( R n ) , n 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 μ of compact support on R 2 with the following property: (P) The only function f C ( R 2 ) satisfying R 2 f σ d μ = 0 for all rigid motions σ of 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.

How to cite

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Brown, L., Schreiber, B., and Taylor, B. A.. "Spectral synthesis and the Pompeiu problem." Annales de l'institut Fourier 23.3 (1973): 125-154. <http://eudml.org/doc/74135>.

@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 -

References

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  12. [12] B. MALGRANGE, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble), 6 (1955-1956), 271-355. Zbl0071.09002MR19,280a
  13. [13] D. POMPEIU, Sur une propriété des fonctions continues dépendent de plusieurs variables, Bull. Sci. Math. (2), 53 (1929), 328-332. Zbl55.0138.04JFM55.0138.04
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Citations in EuDML Documents

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  1. Yitzhak Weit, On Schwartz's theorem for the motion group
  2. Roger Gay, Inversion de la transformation de Pompeiu locale
  3. Axel Osses, Jean-Pierre Puel, Approximate controllability for a linear model of fluid structure interaction
  4. Axel Osses, Jean-Pierre Puel, Approximate controllability for a linear model of fluid structure interaction
  5. Sundaram Thangavelu, Mean periodic functions on phase space and the Pompeiu problem with a twist
  6. Michael Vogelius, An inverse problem for the equation u = - c u - d
  7. Robert Dalmasso, Le problème de Pompeiu

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