## Generalized localization of Fourier series with respect to the eigenfunctions of the Laplace operator in the classes $$L_ p$$.(English. Russian original)Zbl 0799.35169

Lith. Math. J. 31, No. 3, 269-282 (1991); translation from Lit. Mat. Sb. 31, No. 3, 387-405 (1991).
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 $$p\geq 2$$. 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 and eigenfunction expansions in context of PDEs 42C25 Uniqueness and localization for orthogonal series 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

### Keywords:

generalized localization in $$L_ p$$
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### References:

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