Wavelet bi-frames with few generators from multivariate refinable functions

Using results on syzygy modules over a multivariate polynomial ring, we are able to construct compactly supported wavelet bi-frames with few generators from almost any pair of compactly supported multivariate refinable functions. In our examples, we focus on wavelet bi-frames whose primal and dual wavelets are both derived from one box spline function. Our wavelet bi-frames have fewer generators than comparable constructions available in the literature.