Riesz bases in weighted spaces

A. A. Putintseva

Ufa mathematical journal, Tome 3 (2011) no. 1, pp. 45-50

Résumé

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 a space of locally integrable functions on $I$, such that $$ \|f\|:=\sqrt{\int _I|f(t)|^2e^{-2h(t)}\,dt}\infty $$ by $L_2(I,h)$. If $I=(-\pi;\pi)$, $h(t)\equiv1$, 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. As it has been shown by B. J. Levin, nonharmonic Riesz bases in $L_2(-\pi;\pi)$ can be constructed using a system of zeros of entire functions of sine type. In this paper we prove that if a Riesz basis of exponentials exists in the space $L_2(I,h)$, 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 $L_2(-\pi;\pi)$.

Détail

Keywords: Riesz basis, weighted Hilbert spaces, reproducing kernel, functions оf sine type.
Mots-clés : Fourier–Laplace transform