Real algebraic geometry, moment problems and multivariate tight wavelet frames

Details Real algebraic geometry, moment problems and multivariate tight wavelet frames Real algebraic geometry, moment problems and multivariate tight wavelet frames

2012-02-16

arxiv.org/abs/1202.3596v2

Mathematics

Functional Analysis

Scientific paper

In this paper we employ recent results from real algebraic geometry and theory of moment problems to make the first step towards resolving the question of existence of multivariate tight wavelet frames whose generators have at least one vanishing moment. The so-called Unitary Extension Principle by A. Ron and Z.Shen and the results by M. J. Lai and J. St\"ockler allow us to reformulate the question of existence of tight wavelet frame in terms of the existence of the sum of squares decomposition of a single trigonometric polynomial with real coefficients. Our main result confirms the existence of such decompositions in the two-dimensional case. We also give sufficient conditions for existence of tight wavelet frames in the dimension $d \ge 3$ and illustrate our results with several examples.

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