Quantcast

On the equiconvergence rate of trigonometric series expansions and eigenfunction expansions for the Sturm-Liouville operator with a distributional potential

Research paper by I. V. Sadovnichaya

Indexed on: 19 Jul '08Published on: 19 Jul '08Published in: Differential Equations



Abstract

We study the Sturm-Liouville operator L = −d2/dx2 + q(x) in the space L2[0, π] with the Dirichlet boundary conditions. We assume that the potential has the form q(x) = u′(x), u ∈ W2θ[0, π], 0 < θ < 1/2. We consider the problem on the uniform (on the entire interval [0, π]) equiconvergence of the expansion of a function f(x) in a series in the system of root functions of the operator L with its Fourier expansion in the system of sines. We show that if the antiderivative u(x) of the potential belongs to any of the spaces W2θ[0, π], 0 < θ < 1/2, then the equiconvergence rate can be estimated uniformly over the ball u(x) ∈ BR = {v(x) ∈ W2θ[0, π] | ∥v∥W2θ ≤ R} for any function f(x) ∈ L2[0, π].