Characterizations of Scaling Functions: Continuous Solutions

A dilation equation is a functional equation of the form $f( t ) = \sum_{k = 0}^N {c_k } f ( 2t - k )$, 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 $\{ c_0 , \ldots ,c_N \}$. The arguments lead directly to a characterization of all dilation equations that have continuous, compactly supported solutions.