Sobolev exponents of Butterworth refinable functions

Abstract

The precise Sobolev exponent s(infinity)(phi(n)) of the Butterworth refinable function phi(n) associated with the Butterworth filter of order n, b(n)(xi) := cos(2n)(xi/2)/cos(2n)(xi/2)+sin(2n)(xi/2), is shown to be s(infinity) (phi(n)) = n log(2) 3 + log(2) (1 + 3(-n)). This recovers the previously given asymptotic estimate of s(infinity) (phi(n)) of Fan and Sun, and gives more accurate regularity of Butterworth refinable function phi(n). (C) 2007 Elsevier Ltd. All rights reserved.