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

Bastys, A.

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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