Equiconvergence of Spectral Decompositions for Sturm–Liouville Operators: Triples of Lebesgue Spaces

Abstract

The paper deals with the Sturm–Liouville operator generated on the finite interval $[0,\pi]$ by the differential expression $-y^{\prime\prime}+q(x)y$, where $q=u^{\prime}$, $u\in L_{\varkappa}[0,\pi]$ for some $\varkappa\geq 2$, and arbitrary regular boundary conditions. Consider two such operators with different potentials but the same boundary conditions. We prove that the difference between spectral decompositions $S_{m}^{1}(f)-S_{m}^{2}(f)$ of this operators tends to zero as $m\to\infty$ for any $f\in L_{\mu}[0,\pi]$ in the norm of the space $L_{\nu}[0,\pi]$ if the indices satisfy the inequality $1/\varkappa+1/\mu-1/\nu\leq 1$ (except for the case $\varkappa=\nu=\infty$, $\mu=1$). In particular, in the case of a square summable function $u$ the uniform equiconvergence on the whole interval $[0,\pi]$ is proved for an arbitrary function $f\in L_{2}[0,\pi]$