An Algebraic Perspective on Multivariate Tight Wavelet Frames with Rational Masks.

The ideas of sums of squares representations for polynomials and trigonometric polynomials are classical, but new results and applications for them continue to be discovered. Recently, new tight wavelet frame constructions have been found that use sums of squares representations for certain nonnegative trigonometric polynomials to obtain wavelet masks generating a tight wavelet frame for $L^{2}(\mathbb{R}^{n})$. While there has been some effort to extend these results to obtain tight wavelet frames with higher vanishing moments, which have better approximation properties, this has been done under restrictive assumptions. In this paper, we generalize the previous constructions using sums of squares to allow for the inclusion of a vanishing moment recovery function. This results in rational wavelet masks generating tight wavelet frames for $L^{2}(\mathbb{R}^{n})$, with lower bounds on their numbers of vanishing moments. In particular, our construction may be used to create tight wavelet frames with better vanishing moments in a wide variety of new settings. Our construction is made possible by a new theoretical result, which states that nonnegative multivariate trigonometric polynomials always have representations as sums of squares of rational functions. This theorem settles a long-standing question in the area of sums of squares representations for trigonometric polynomials.