On a class of biorthogonal expansions in exponential functions

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.
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