Type of Publication:  Preprint 
Author:  Charina, Maria; Putinar, Mihai; Scheiderer, Claus; Stöckler, Joachim 
Year of publication:  2012 
ArXivID:  arXiv:1202.3596 
Summary: 
Recent results from real algebraic geometry and the theory of polynomial optimization are related in a new framework to the existence question of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the socalled Unitary Extension Principle by Ron and Shen are interpreted in terms of hermitian sums of squares of certain nonnegative trigonometric polynomials and in terms of semidefinite programming. The latter together with the recent results in algebraic geometry and semidefinite programming allow us to answer affirmatively the long standing open question of the existence of such tight wavelet frames in dimension $d=2$; we also provide numerically efficient methods for checking their existence and actual construction in any dimension. We exhibit a class of counterexamples in dimension $d=3$ showing that, in general, the UEP property is not sufficient for the existence of tight wavelet frames. On the other hand we provide stronger sufficient conditions for the existence of tight wavelet frames in dimension $d > 3$ and illustrate our results by several examples.

Subject (DDC):  510 Mathematics 
Bibliography of Konstanz:  Yes 
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CHARINA, Maria, Mihai PUTINAR, Claus SCHEIDERER, Joachim STÖCKLER, 2012. A real algebra perspective on multivariate tight wavelet frames
@unpublished{Charina2012algeb23509, title={A real algebra perspective on multivariate tight wavelet frames}, year={2012}, author={Charina, Maria and Putinar, Mihai and Scheiderer, Claus and Stöckler, Joachim} }