Sobolev exponents of Butterworth refinable functions

The precise Sobolev exponent s∞(φn)s∞(φn) of the Butterworth refinable function φnφn associated with the Butterworth filter of order nn, bn(ξ)≔cos2n(ξ/2)cos2n(ξ/2)+sin2n(ξ/2), is shown to be s∞(φn)=nlog23+log2(1+3−n)s∞(φn)=nlog23+log2(1+3−n). This recovers the previously given asymptotic estimate of s∞(φn)s∞(φn) of Fan and Sun, and gives more accurate regularity of Butterworth refinable function φnφn.