An algebraic perspective on multivariate tight wavelet frames
Using results of real algebraic geometry, the authors investigate the existence and construction of \(d\)-variate tight wavelet frames whose generators have at least one vanishing moment. The main results of the interesting paper are formulated as polynomial identities or polynomial inequalities. First, equivalent formulations of the unitary extension principle introduced by \textit{A. Ron} and \textit{Z. Shen} [J. Funct. Anal. 148, No. 2, 408--447 (1997; Zbl 0891.42018)] are given in terms of Hermitian sums of squares of certain nonnegative Laurent polynomials and in terms of semidefinite programming. These formulations lead to an affirmative answer to the question of the existence of tight wavelet frames in dimension \(d=2\). For \(d\geq 2\), the results provide efficient numerical methods for checking the existence of tight wavelet frames and for their construction. The authors show that for \(d=3\), the so-called sub-QMF condition is not sufficient for the existence of tight wavelet frames. Therefore, stronger sufficient conditions for the existence of tight wavelet frames are presented for \(d\geq 3\).
numerical construction of tight wavelet frames
scientific article; zbMATH DE number 6225222
scientific article