Total positivity and refinable functions with general dilation

Abstract

We show that a refinable function φ with dilation M⩾2 is a ripplet, i.e., the collocation matrices of its shifts are totally positive, provided that the symbol p of its refinement mask satisfies certain conditions. The main condition is that p (of degree n) satisfies what we term condition (I), which requires that n determinants of the coefficients of p are positive and generalises the conditions of Hurwitz for a polynomial to have all negative zeros. We also generalise a result of Kemperman to show that (I) is equivalent to an M-slanted matrix of the coefficients of p being totally positive. Under condition (I), the ripplet φ satisfies a generalisation of the Schoenberg–Whitney conditions provided that n is an integer multiple of M−1. Moreover, (I) implies that polynomials in a polyphase decomposition of p have interlacing negative zeros, and under these weaker conditions we show that φ still enjoys certain total positivity properties.