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Eigenfunction expansions for generalized functions of several variables

Lamb, W. and McGhee, D.F. (2004) Eigenfunction expansions for generalized functions of several variables. Integral Transforms and Special Functions, 15 (3). pp. 239-249. ISSN 1065-2469

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Abstract

The constructive method developed by Zemanian [Zemanian, A. H. (1968). Generalized Integral Transformations. Interscience, New York] for extending L2-convergence results on eigenfunction expansions to certain classes of generalized functions of one variable is shown to be valid also for generalized functions of several variables. In the latter case, the expansions involve the eigenfunctions associated with symmetric partial differential operators. Specific examples considered are the Laplace-Beltrami operator on the unit sphere in ℝN and a class of symmetric elliptic operators in L2(Φ#169;), where Φ#169; is a bounded region in ℝN. Applications to the solution of distributional initial-boundary value problems are also discussed.