Convergence problems of the eigenfunction expansions for polyharmonic operators

This research focuses on convergence and summability problems of the eigenfunctions expansions of differential operators related to polyharmonic operator in closed domain. The polyharmonic operator (-- ∆)‴,m ∈ Z+ is the elliptic operator of order 2m with domain consists of classes of infinitely differentiable functions with compact support, which is a symmetric and nonnegative linear operator and has a self-adjoint extension. For domains with smooth boundary, the solution to these differential operator problems involves eigenfunction expansions associated with polyharmonic operator with Navier boundary conditions. Suitable estimations for spectral function of the polyharmonic operator by using the mean value formula for the eigenfunctions of the polyharmonic operator is established. These estimations enable us to show the uniformly convergence of the Riesz means of the spectral expansions related to polyharmonic operator in closed domain. The classes of differentiable functions used are Sobolev and Nikolskii classes. Subsequently, the results are applied to study the sufficient conditions for localization properties of the spectral expansions related to distributions. The conditions and principles for the localization of the Riesz means spectral expansions of distributions associated with the polyharmonic operator in closed domain are considered.

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