# Multivariate Tight Wavelet Frames with Few Generators and High Vanishing Moments

## TL;DR

This paper introduces a new method for constructing multivariate tight wavelet frames that use few generators and achieve high vanishing moments, especially for box splines, improving upon existing methods.

## Contribution

The authors develop a general approach combining sums of squares with wavelet frame construction, enabling high vanishing moments with fewer wavelet masks for box splines.

## Key findings

- Achieved high vanishing moments for all wavelet masks.
- Matched the upper bound on the number of highpass masks.
- Demonstrated improved wavelet frame constructions for box splines.

## Abstract

Tight wavelet frames are computationally and theoretically attractive, but most existing multivariate constructions have various drawbacks, including low vanishing moments for the wavelets, or a large number of wavelet masks. We further develop existing work combining sums of squares representations with tight wavelet frame construction, and present a new and general method for constructing such frames. Focusing on the case of box splines, we also demonstrate how the flexibility of our approach can lead to tight wavelet frames with high numbers of vanishing moments for all of the wavelet masks, while still having few highpass masks: in fact, we match the best known upper bound on the number of highpass masks for general box spline tight wavelet frame constructions, while typically achieving much better vanishing moments for all of the wavelet masks, proving a nontrivial lower bound on this quantity.