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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 $p\ge 2$. 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:
 35P10 Completeness of eigenfunctions, eigenfunction expansions for PD operators 42C25 Uniqueness and localization for orthogonal series 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
##### References:
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