D. Colella et C. Heil, CHARACTERIZATIONS OF SCALING FUNCTIONS - CONTINUOUS SOLUTIONS, SIAM journal on matrix analysis and applications, 15(2), 1994, pp. 496-518
A dilation equation is a functional equation of the form f(t) = SIGMA(
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 i
n several fields, including interpolating subdivision schemes and wave
let theory. This paper obtains sharp bounds for the Holder exponent of
continuity of any continuous, compactly supported scaling function in
terms of the joint spectral radius of two matrices determined by the
coefficients {c0, ..., C(N)}. The arguments lead directly to a charact
erization of all dilation equations that have continuous, compactly su
pported solutions.