Summary
We consider the variational solution of a particular class of second order differential equations and show that expansions in terms of Chebychev and a range of ultraspherical polynomials lead to operator matrices that are asymptotically diagonal, and that hence their convergence properties can be completely characterised using a previously developed analysis. For a given class of weight functions bounds are given on the convergence of the coefficients and of the weighted mean square error, in terms of the analyticity properties of the coefficients in the differential equation. These bounds are used to discuss the optimum choice of weight function for such a calculation.
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Delves, L.M., Bain, M. On the optimum choice of weight functions in a class of variational calculations. Numer. Math. 27, 209–218 (1977). https://doi.org/10.1007/BF01396641
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DOI: https://doi.org/10.1007/BF01396641