Pointwise and L(p)(IRn) Convergence of Elliptic Eigenfunction Expansions

摘要

This paper presents some results on equisummability and regularization of eigenfunction expansions associated with a general class of elliptic operators on (r sub n). The prototypical analogue of equisummability is equiconvergence. Equiconvergence asks: When does the pointwise convergence of an eignefunction expansion associated with an unperturbed operator (say the Laplacian) imply the pointwise convergence of the perturbed operator (say the Laplacian plus non-zero potential). The problem of equisummability is essentially the question of convergence to zero of the difference of the two summability means of the unperturbed and perturbed eigenfunction expansions. The main equisummability result are applied to singular Sturm Liouville theory and the heat equation. Regularization of the ill posed problem for partial differential equations where the solutions do not depend continuously on the data is presented in the latter part of this paper. That is, a stable method is developed for correct summation of expansions of data perturbed by error. The method recovers from the perturbed expansion a good approximation to the correct data provided the data are sufficiently regular. Application to the study of time evolution of the Laplacian is mentioned. (Reprints)