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

Abstract:

Given the Sturm-Liouville eigenfunction expansion of an ${L_2}$ function $f(x)$, summability theory provides means for recovering the value of $f({x_0})$ at points ${x_0}$ where $f$ is sufficiently regular. If the coefficients in the expansion are perturbed slightly (in the ${L_2}$ norm), a stable summation method will recover from the perturbed expansion a good approximation to $f({x_0})$. In this paper we develop stable summation methods for expansions in eigenfunctions of the singular Sturm-Liouville system $u'' - q(x)u = - \lambda u,u(0)\cos \beta + u’(0)\sin \beta = 0,u(\infty ) < \infty$; where $q(x)$ and continuous. Given a summability method known to work at ${x_0}$ for a particular expansion, our results say that if the summation parameter is appropriately scaled with the ${L_2}$ 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.