Dyadic wavelets and refinable functions on a half-line

Abstract

For an arbitrary positive integer refinable functions on the positive half-line are defined, with masks that are Walsh polynomials of order . The Strang-Fix conditions, the partition of unity property, the linear independence, the stability, and the orthonormality of integer translates of a solution of the corresponding refinement equations are studied. Necessary and sufficient conditions ensuring that these solutions generate multiresolution analysis in are deduced. This characterizes all systems of dyadic compactly supported wavelets on and gives one an algorithm for the construction of such systems. A method for finding estimates for the exponents of regularity of refinable functions is presented, which leads to sharp estimates in the case of small . In particular, all the dyadic entire compactly supported refinable functions on are characterized. It is shown that a refinable function is either dyadic entire or has a finite exponent of regularity, which, moreover, has effective upper estimates.