<title>Characterization of polynomials satisfying the super-Haar condition</title>

Abstract

The scaling function for the Super Haar wavelet is a linear combination of shifts in the Haar scaling function; the coefficients of this linear combination are assumed to be integers. If the scaling function satisfies the dilation equation the coefficients are said to be Super Haar Admissible. It has been shown that the z transform of Super Haar Admissible coefficients results in a polynomial that satisfies certain conditions. We define a related condition, which we call the Super Haar Condition and show that cyclotomic polynomials of odd order satisfy it. Further, dilation coefficients associated with such polynomials can immediately be found from relations among the cyclotomic polynomials. Using these results, a large class of Super Haar Admissible coefficients is identified and we conjecture that this class includes all admissible coefficients. We discuss applications to denoising and present an example.