Cesàro-one summability and uniform convergence of solutions of a Sturm-Liouville system (Q817093)

Let \(f\) be a du Bois-Reymond function on \([0, \pi]\) with \(f(0) = f(\pi) = 0\), the Fourier series of \(f\) is \(\sum_{n=1}^\infty \alpha_n \phi_n\) with \(\phi_n (x) = \sin nx\). It is known that the \((C, 1)\) sum of \(y_N = \sum_{n=1}^N \alpha_N \phi_N\) converges uniformly to \(f\) on \([0, \pi]\). The authors view \(y_N\) as a solution of the Sturm-Liouville problem \[ y^{\prime \prime} + y = g_N,\;y(0) = y(\pi) = 0, \] with \(g_N = \sum_{n=1}^N \lambda_n \alpha_n \phi_n, \lambda_n = 1 - n^2\). In this sense, every continuous function on \([0, \pi]\), after a slight modification to fit the boundary condition, is the uniform limit of solutions (in the Galerkin type) of nonhomogeneous Sturm-Liouville problems.