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 L2(I,h) a space of locally integrable functions on I, such that ||f||:=√∫I|f(t)|2e−2h(t)dt<∞. In the case where I=(− pi; pi), h(t) equiv1, the space L2(I,h) coincides with the classical space L2(− pi; pi) and the Fourier trigonometric system is a Riesz basis in this space. Nonharmonic Riesz bases in L2(− 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 L2(I,h) exists a Riesz basis of exponentials, this space is isomorphic (as a normed space) to the classical space L2(I). Thus, the existence of Riesz bases of exponentials is the exclusive property of the classical space L2(− pi; pi).