Extension of convergence of quasipolynomials

A. M. Sedletskii

Izvestiya. Mathematics , Tome 17 (1981) no. 2, pp. 353-368

Résumé

The system $\{\exp(i\lambda_nx)\}$, minimal in $L^p(-a,a)$ ($a\infty$, $1\leqslant p\leqslant\infty$), is called a system of extension of $L^p$-convergence if any sequence of linear combinations of this system converging in $L^p(-a,a)$ converges in $L^p$-norm on every finite interval. A complete description of systems of extension of convergence is given in the class of systems $\{\exp(i\lambda_nx)\}$ generated by sequences of zeros of entire functions of the form $$ L(z)=\int_{-a}^a \frac{e^{izt}k(t)}{(a-|t|)^\alpha}\,dt,\quad0\alpha1,\quad\operatorname{var}k(t)\infty,\quad k(\pm a\mp0)\ne0, $$ where $k(t)$ has, in addition, a certain smoothness in a neighborhood of the points $\pm a$. Specifically, for $1$ this property is realized if and only if $\alpha\ne1-1/p$, while for $p=1$ or $\infty$ there is no extension of convergence. This result is applied to the question of bases of exponential functions in $L^p(-a,a)$, $1$. Bibliography: 13 titles.