Equiconvergence of spectral decompositions for the Dirac system with potential in Lebesgue spaces

I. V. Sadovnichaya

Informatics and Automation, Function spaces, approximation theory, and related problems of mathematical analysis, Tome 293 (2016), pp. 296-324

Résumé

The problem of equiconvergence of spectral decompositions corresponding to the systems of root functions of two one-dimensional Dirac operators $\mathcal L_{P,U}$ and $\mathcal L_{0,U}$ with potential $P$ summable on a finite interval and Birkhoff-regular boundary conditions is analyzed. It is proved that in the case of $P\in L_\varkappa[0,\pi]$, $\varkappa\in(1,\infty]$, equiconvergence holds for every function $\mathbf f\in L_\mu[0,\pi]$, $\mu\in[1,\infty]$, in the norm of the space $L_\nu[0,\pi]$, $\nu\in[1,\infty]$, if the indices $\varkappa,\mu$, and $\nu$ satisfy the inequality $1/\varkappa+1/\mu-1/\nu\le1$ (except for the case when $\varkappa=\nu=\infty$ and $\mu=1$). In particular, in the case of a square summable potential, the uniform equiconvergence on the interval $[0,\pi]$ is proved for an arbitrary function $\mathbf f\in L_2[0,\pi]$.