On refinement masks of tight wavelet frames

TL;DR

This paper establishes conditions under which trigonometric polynomials can serve as refinement masks for tight wavelet frames, linking the roots of associated algebraic polynomials to the mask's properties.

Contribution

It provides new sufficient conditions based on roots of algebraic polynomials for a trigonometric polynomial to be a refinement mask of a tight wavelet frame.

Findings

Any trigonometric polynomial with an associated algebraic polynomial having only negative roots can serve as a refinement mask.

The paper characterizes when a polynomial can be a refinement mask based on root location.

Conditions are given for the existence of tight wavelet frames from given masks.

Abstract

In the paper we obtain sufficient conditions for a trigonometric polynomial to be a refinement mask corresponding to a tight wavelet frame. The condition is formulated in terms of the roots of a mask. In particular, it is proved that any trigonometric polynomial can serve as a mask if its associated algebraic polynomial has only negative roots (at least one of them, of course, equals $-1$).