Total positivity and refinable functions with general dilation
Abbreviated Journal Title
Appl. Comput. Harmon. Anal.
total positivity; zeros of polynomials; refinable function; wavelet; SCALING FUNCTIONS; WAVELETS; CONSTRUCTION; EQUATIONS; MATRIX; Mathematics, Applied; Physics, Mathematical
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.
Applied and Computational Harmonic Analysis
"Total positivity and refinable functions with general dilation" (2004). Faculty Bibliography 2000s. 4383.