On a~class of biorthogonal expansions in exponential functions

A. M. Sedletskii

Izvestiya. Mathematics , Tome 11 (1977) no. 2, pp. 375-395

Résumé

We consider a biorthogonal expansion in terms of the system $\{e^{\lambda_nx}\}$, where $\lambda_n$ are the zeros of the entire function $$ L(z)=h_0e^z+\int_0^1e^{zt}k(t)\,dt,\qquad h_0\ne0, $$ and $k^{(m)}(t)$ has bounded variation for some integer $m\geqslant0$, $k^{(j)}(0)=0$ for $j=0,1,\dots,m-1$ and $k^{(m)}(0+0)\ne0$. The function to be expanded has domain $(0,1)$. We describe the sets of convergence (and divergence) of the series for the classes $L^p$, $C$, $\operatorname{Lip}\alpha$, and $V$. The results indicate that the series have properties different from those of ordinary Fourier series; and the difference becomes more pronounced as $m$ increases. Bibliography: 16 titles.