ON SOME SHARP REGULARITY ESTIMATIONS OF L(2)-SCALING FUNCTIONS

Let f be a compactly supported L(2)-solution of the two-scale dilation equation and alpha be the L(2)-Lipschitz exponent of f. We prove, in addition to other results, that there exists an integer k greater than or equal to 0 such that (i) 1/h(2 alpha)\ln h\(k)) integral(-infinity )(infinity)\f(x + h) - f(x)\(2)dx approximate to p(h) as h --> 0(+), w here p is a nonzero bounded continuous function with p(2h) = p(h), and (ii) for s > alpha, there exists a nonzero bounded continuous q (depe nds on s) with q(2T) = q(T) and 1/T-2(s-alpha)(ln T)(k) integral(-T)(T ) \omega(s) (f) over cap(omega)\(2)d omega approximate to) q(T) as T - -> infinity. The above alpha and k can be calculated through a transit ion matrix. These improve the previous result of Cohen and Daubechies concerning the Besov space containing f and Villemoes's result on the Sobolev exponent of (f) over cap.