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.


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
Full Text: DOI


[1] L. G?rding, ?Eigenfunction expansions connected with elliptic differential operators,? in: Tolfte Skandinaviska Matematikerkongressen, Lund, 1953, pp. 44-55 (1954).
[2] V. A. Il’in, ?On a generalized interpretation of the principle of localization for Fourier series with respect to fundamental systems of functions,? Sib. Mat. Zh.,9, No. 5, 1093-1106 (1968). · Zbl 0191.07503
[3] P. Sj?lin, ?Regularity and integrability of spherical means,? Monatsh. Math.,96, No. 4, 277-291 (1983). · Zbl 0519.42018
[4] A. J. Bastys [A. Bastis], ?The generalized localization principle for an N-fold Fourier integral,? Dokl. Akad. Nauk SSSR,278, No. 4, 777-778 (1984).
[5] K. I. Babenko, ?On the summability and the convergence of the expansions in the eigenfunctions of a differential operator,? Mat. Sb.,91, No. 2, 147-201 (1973).
[6] Sh. A. Alimov, V. A. Il’in, and E. M. Nikishin, ?Questions on the convergence of multiple trigonometric series and spectral expansions. I,? Usp. Mat. Nauk,31, No. 6, 28-82 (1976).
[7] A. J. Bastys [A. Bastis], ?On the generalized localization principle for an N-fold Fourier integral in the classes Lp,? Dokl. Akad. Nauk SSSR,304, No. 3, 526-529 (1989).
[8] C. Fefferman, ?The multiplier problem for the ball,? Ann. Math.,94, No. 2, 330-336 (1971). · Zbl 0234.42009
[9] H. Busemann and W. Feller, Differentiation der L-Integrale,? Fund. Math.,22, 226-256 (1934). · JFM 60.0218.03
[10] A. Zygmund, Trigonometric Series, Vols. I and II, Cambridge Univ. Press, Cambridge (1959). · Zbl 0085.05601
[11] V. A. Il’in, ?On the expansion of functions with singularities into a conditionally convergent series of eigenfunctions,? Izv. Akad. Nauk SSSR, Ser. Mat.,22, 49-80 (1958).
[12] G. N. Watson, A Treatise on the Theory of Bessel Functions, Macmillan, New York (1944). · Zbl 0063.08184
[13] Handbook of Special Functions [in Russian], Nauka, Moscow (1979).
[14] E. M. Nikishin, ?Resonance theorems and functional series,? Doctoral Dissertation, Moscow State University, Mowcow (1971).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.