Characterizations of Scaling Functions: Continuous Solutions *

A dilation equation is a functional equation of the form f (t) = N k=0 c k f (2t − k), and any nonzero solution of such an equation is called a scaling function. Dilation equations play an important role in several fields, including interpolating subdivision schemes and wavelet theory. This paper obtains sharp bounds for the Hölder exponent of continuity of any continuous, compactly supported scaling function in terms of the joint spectral radius of two matrices determined by the coefficients {c 0 ,. .. , c N }. The arguments lead directly to a characterization of all dilation equations that have continuous, compactly supported solutions. Functional equations of the form (1) f (t) = N k=0 c k f (2t − k) play an important role in several fields, including wavelet theory and interpolating subdivision schemes. Such equations are referred to as dilation equations or two-scale difference equations, and any nonzero solution f is called a scaling function. The coefficients {c 0 ,. .. , c N } may be real or complex; if they are real then the scaling function f will be real-valued. In this paper we obtain sharp bounds for the Hölder exponent of continuity of any continuous, compactly supported scaling function. Our arguments lead directly to a characterization of all dilation equations that have continuous, compactly supported solutions. These methods also enable us to examine how certain properties of scaling functions, such as the Hölder exponent, behave as a function of the coefficients, and we provide several examples to illustrate the basic structure present. Our work was inspired by an early preprint of [DL2], in which sufficient conditions for the existence of continuous, compactly supported scaling functions were obtained and lower bounds for the Hölder exponent of continuity were derived. In that paper the assumption was made that the coefficients satisfy (2) k c 2k = k c 2k+1 = 1. Conditions and bounds were then expressed in terms of the joint spectral radiusˆρ(T 0 | V , T 1 | V) of two matrices T 0 , T 1 (determined by the coefficients {c 0 ,. .. , c N }) restricted to a certain subspace V of C N. We have extended these results in the *