Abstract:
We obtain a result concerning the basis property
in a weighted space on an interval $(-a,a)$
for a system of exponentials generated by the zeros
of the Fourier transform of a function with singularities
at the ends of the support interval $(-a,a)$.
For an arbitrary $\Delta\in\mathbb{C}$ we find a criterion
for the basis property of the system
$(e^{i(n+\Delta\operatorname{sign} n)t})_{n\in\mathbb{Z}}$
in a weighted space on the interval $(-\pi,\pi)$ and the systems
of sines $(\sin((n+\Delta)t))_{n\in\mathbb{N}}$ and cosines
$1\cup (\cos((n+\Delta)t))_{n\in\mathbb{N}}$ in a weighted
space on the interval $(0,\pi)$. The weight is everywhere
a finite product of polynomial functions.
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