A note on the rate of convergence of Sturm-Liouville expansions (Q1822251)

Let \((g_ k)\) be the sequence of orthonormalized eigenfunctions of a given linear Sturm-Liouville problem. Let \(S_ n(f,x)\) be the nth partial sum of the eigenfunction expansion (in terms of the \(g_ k)\) of a function f of bounded variation on (0,\(\pi)\). The author shows that the difference \((f(x+0)+f(x-0))-S_ n(f,x)\) is bounded by an expression involving a constant and the total variation of f. The approach is elementary and uses only properties of asymptotic expansion of eigenfunctions and eigenvalues.