RIESZ BASES IN WEIGHTED SPACES

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 parallel to f parallel to := root integral(I) vertical bar f(t)vertical bar(2)e(-2h(t)) dt < infinity by L-2 (I, h). If I = (-pi; pi), h(t) 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. 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 a 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).