Article
Ufa Mathematical Journal
Volume 3, Number 1, pp. 45-50
Riesz bases in weighted spaces.
Putintseva A.A.
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The article deals with weighted Hilbert spaces with convex weights. Let $ h $ be a convex function on a bounded interval $ I $ of the real axis. We denote by $ L_2 (I, h) $ a space of locally integrable functions on $ I $, such that $$ ||f||:=\sqrt {\int _I |f(t)|^2e^{-2h(t)}\,dt} <\infty . $$ In the case where $ I = (- \ pi; \ pi) $, $ h (t) \ equiv 1 $, the space $ L_2 (I, h) $ coincides with the classical space $ L_2 (- \ pi; \ pi) $ and the Fourier trigonometric system is a Riesz basis in this space. Nonharmonic Riesz bases in $ L_2 (- \ pi; \ pi) $, as shown in B.J. Levin, can be designed using a system of zeros of entire functions of sine type. In this paper we prove that if in the space $ L_2 (I, h) $ exists a Riesz basis of exponentials, this space is isomorphic (as a normed space) to the classical space $ L_2 (I) .$ Thus, the existence of Riesz bases of exponentials is the exclusive property of the classical space $ L_2 (- \ pi; \ pi) $.