Stable Summation Methods for a Class of Singular Sturm-liouville Expansions

Given the Sturm-Liouville eigenfunction expansion of an ¿2 function fix), summability theory provides means for recovering the value of fix¿) at points x0 where / is sufficiently regular. If the coefficients in the expansion are perturbed slightly (in the I^ norm), a stable summation method will recover from the perturbed expansion a good approximation to fix¿). In this paper we develop stable summation methods for expansions in eigenfunctions of the singular Sturm-Liouville system u" — q(x)u = -\u, n(0)cos ß + u'(0)sin ß « 0, u(oo) < oo; where q(x) e L,[0, oo) and continuous. Given a summability method known to work at x0 for a particular expansion, our results say that if the summation parameter is appropriately scaled with the L^ 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.