, Volume 1, Issue 3, pp 249-279

Tight Frames of Polynomials and the Truncated Trigonometric Moment Problem

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Abstract

A simple parametrization is given for the set of positive measures with finite support on the circle group T that are solutions of the truncated trigonometric moment problem: \(\hat{\mu}(k)=s_k, |k|\le N,\) where the parameters are, up to nonzero multiplicative constants, the polynomials whose roots all have a modulus less than one. This result is then used to characterize, on a certain natural Hilbert space of polynomials associated with the problem, all finite "weighted" tight frames of evaluation polynomials. Finally, a new and simple way of parametrizing the whole set of positive Borel measures on T, solutions of the given moment problem is deduced, via a limiting argument.