## Generating Tight Wavelet Frames From Sums of Squares Representations

##### Abstract

We construct multivariate tight wavelet frames in several settings by using the theory of sums of squares representations for nonnegative trigonometric polynomials. This is done by way of two extension principles which allow us to translate the problem of constructing these frames to one of designing collections of trigonometric polynomials satisfying certain orthogonality and normalization conditions. We consider first the setting of dyadic dilation, and assume that the lowpass masks are constructed by the coset sum method, which lifts a univariate lowpass mask to a nonseparable multivariate lowpass mask, with several properties of the input being preserved. The existence of the necessary sums of squares representations is proved utilizing the special structure of these lowpass masks. We extend this first construction to the setting of prime dilation, focusing on the case of interpolatory input masks. We prove lower bounds on the vanishing moments of the highpass masks in these two constructions, and new results about the properties of the prime coset sum method.
In the first two settings, we use lowpass masks satisfying the sub-QMF condition, and apply the unitary extension principle to ensure that our filter banks result in tight wavelet frames. In the third setting, we use lowpass masks satisfying a generalization of this condition, which we dub the oblique sub-QMF condition. In fact, it turns out that for a fixed lowpass mask and vanishing moment recovery function, this condition is equivalent to the existence of highpass masks satisfying the oblique extension principle conditions. This allows us to construct multivariate tight wavelet frames for any lowpass mask satisfying the oblique sub-QMF condition, under some mild assumptions on the vanishing moment recovery function. To establish this equivalence, we first prove a new result on sums of squares representations for nonnegative multivariate trigonometric polynomials, which says that any such function may be written as a finite sum of squares of quotients of trigonometric polynomials.
We will also prove a generalization of the sum of squares result for matrices rather than functions, in which we show that a matrix with trigonometric polynomial entries which is positive semidefinite for all evaluations has a representation as a sum of squares of commuting symmetric matrices with rational trigonometric polynomial entries. We suspect that these sums of squares results for trigonometric polynomials and matrices with such entries will be of interest far beyond the wavelet construction community.