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Generalized localization of Fourier series with respect to the eigenfunctions of the Laplace operator in the classes L p . (English) Zbl 0799.35169
Summary: The article deals with the problem of localization almost everywhere of Fourier series with eigenfunctions of the Laplace operator in L p - spaces. It is proved that the necessary and sufficient condition for the generalized localization in L p -spaces for many-fold Fourier integral, summarized by sphere, is p2. For p<2, it is proved that for every L 2 selfadjoint extension of the Laplace operator there exists a function f in L p , the spectral decomposition of which diverges in A with measure A>0, although the function f is equal to zero on A. The estimations from below for the norm of the projector of every selfadjoint extension of the Laplace operator in L p -classes are obtained.
MSC:
35P10Completeness of eigenfunctions, eigenfunction expansions for PD operators
42C25Uniqueness and localization for orthogonal series
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
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