Title

Total Positivity And Refinable Functions With General Dilation

Keywords

Refinable function; Total positivity; Wavelet; Zeros of polynomials

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. © 2004 Elsevier Inc.

Publication Date

1-1-2004

Publication Title

Applied and Computational Harmonic Analysis

Volume

16

Issue

2

Number of Pages

69-89

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.acha.2004.01.001

Socpus ID

1542607316 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/1542607316

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