Stable summation methods for a class of singular Sturm-Liouville expansions

Abstract

Given the Sturm-Liouville eigenfunction expansion of an L 2 {L_2} function f ( x ) f(x) , summability theory provides means for recovering the value of f ( x 0 ) f({x_0}) at points x 0 {x_0} where f f is sufficiently regular. If the coefficients in the expansion are perturbed slightly (in the L 2 {L_2} norm), a stable summation method will recover from the perturbed expansion a good approximation to f ( x 0 ) f({x_0}) . In this paper we develop stable summation methods for expansions in eigenfunctions of the singular Sturm-Liouville system u − q ( x ) u = − λ u , u ( 0 ) cos ⁡ β + u ′ ( 0 ) sin ⁡ β = 0 , u ( ∞ ) > ∞ u - q(x)u = - \lambda u,u(0)\cos \beta + u’(0)\sin \beta = 0,u(\infty ) > \infty ; where q ( x ) q(x) and continuous. Given a summability method known to work at x 0 {x_0} for a particular expansion, our results say that if the summation parameter is appropriately scaled with the L 2 {L_2} error in the perturbed expansion, a stable summation method is obtained. We obtain a sharp scaling requirement for guaranteeing stability. We apply our results to Riesz and Stieltjes summability.