Estimates for the convergence rate of spectral decompositions of even-order differential operators

Abstract

This paper studies spectral properties of ordinary differential operators of even order 2n, n > 1, with nonsmooth coefficients in the differential operation, in particular, with nonsmooth coefficient multiplying the (2n – 1)th derivative. We use the Il’in’s approach [1] to studying spectral properties of operators irre� spective of the particular form of the boundary condi� tions. We investigate the dependence of estimates for the rate of local convergence of spectral decomposi� tions on the distance from an interior compact set K to the boundary ∂G in the case where the normalized Fourier coefficients αkfk have asymptotics O () , where ν = const > 0 and λk is the spectral parameter (see Theorem 1) and in the case where the coefficients have asymptotics O (l n –β λ k ), β = const (see Theo� rem 2). We begin with a brief review of related results. Paper [2] studies the rate of convergence of biorthog� onal expansions of functions in systems of root func�