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

Authors: Harvey Diamond, Mark Kon and Louise Raphael
Journal: Proc. Amer. Math. Soc. 81 (1981), 279-286
MSC: Primary 34B25
MathSciNet review: 593472
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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.

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Keywords: Sturm-Liouville expansions, stable summability, singular
Article copyright: © Copyright 1981 American Mathematical Society