Riesz bases generated by the spectra of Sturm-Liouville problems

Abstract

Let $\{\lambda_n^2\}_{n = 0}^\infty$ be the spectra of a Sturm-Liouville problem on $[0,\pi ]$. We investigate the question: Do the systems $\{\cos(\lambda_n x)\}_{n= 0}^\infty$ or $\{\sin(\lambda_n x)\}_{n = 0}^\infty$ form Riesz bases in ${L^2}[0,\pi]$? The answer is almost always positive.