Total positivity and refinable functions with general dilation

    Authors

    T. N. T. Goodman;Q. Y. Sun

    Comments

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    Abbreviated Journal Title

    Appl. Comput. Harmon. Anal.

    Keywords

    total positivity; zeros of polynomials; refinable function; wavelet; SCALING FUNCTIONS; WAVELETS; CONSTRUCTION; EQUATIONS; MATRIX; Mathematics, Applied; Physics, Mathematical

    Abstract

    We show that a refinable function phi with dilation M greater than or equal to 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 (1), 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 (1) is equivalent to an M-slanted matrix of the coefficients of p being totally positive. Under condition (1), the ripplet phi 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 0 still enjoys certain total positivity properties. (C) 2004 Elsevier Inc. All rights reserved.

    Journal Title

    Applied and Computational Harmonic Analysis

    Volume

    16

    Issue/Number

    2

    Publication Date

    1-1-2004

    Document Type

    Article

    Language

    English

    First Page

    69

    Last Page

    89

    WOS Identifier

    WOS:000220003300001

    ISSN

    1063-5203

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