We present two methods for constructing new nonseparable multidimensional tight wavelet frames by combining the ideas of sum of squares representations of nonnegative trigonometric polynomials with the coset sum method of generating nonseparable multidimensional lowpass filters from univariate lowpass filters. In effect, these methods allow one to select a univariate lowpass filter and generate nonseparable multidimensional tight wavelet frames from it in any dimension n≥2, under certain conditions on the input filter which are given explicitly. We construct sum of hermitian squares representations for a particular class of trigonometric polynomials f in several variables, each related to a coset sum generated lowpass mask τ in that nonnegativity of f implies the sub-QMF condition for τ, in two ways: for interpolatory inputs to the coset sum method satisfying the univariate sub-QMF condition, we find this representation using the Fejér–Riesz Lemma; and in the general case, by writing f=x⁎Px, where x is a vector of complex exponential functions, and P is a constant positive semidefinite matrix that is constructed to reduce the number of generators in this representation. The generators of this representation of f may then be used to generate the filters in a tight wavelet frame with lowpass mask τ. Several examples of these representations and the corresponding frames are given throughout.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics