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Spectral synthesis and the Pompeiu problem

L. Brown, B. Schreiber, B. A. Taylor (1973)

Annales de l'institut Fourier

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....

Stratified Whitney jets and tempered ultradistributions on the subanalytic site

N. Honda, G. Morando (2011)

Bulletin de la Société Mathématique de France

In this paper we introduce the sheaf of stratified Whitney jets of Gevrey order on the subanalytic site relative to a real analytic manifold X . Then, we define stratified ultradistributions of Beurling and Roumieu type on X . In the end, by means of stratified ultradistributions, we define tempered-stratified ultradistributions and we prove two results. First, if X is a real surface, the tempered-stratified ultradistributions define a sheaf on the subanalytic site relative to X . Second, the tempered-stratified...

Strong boundary values : independence of the defining function and spaces of test functions

Jean-Pierre Rosay, Edgar Lee Stout (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

The notion of “strong boundary values” was introduced by the authors in the local theory of hyperfunction boundary values (boundary values of functions with unrestricted growth, not necessarily solutions of a PDE). In this paper two points are clarified, at least in the global setting (compact boundaries): independence with respect to the defining function that defines the boundary, and the spaces of test functions to be used. The proofs rely crucially on simple results in spectral asymptotics.

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