A new technique to estimate the regularity of refinable functions (Q1923689)

The regularity of refinable functions satisfying \[ \varphi(x)= 2\sum_n c_n \varphi(2x-n) \] is studied. The regularity of \(\varphi\) is measured by estimating the \(L^p\)-Sobolev exponent \[ s_p= \sup\Biggl\{\gamma: \int|\widehat{\varphi}(\omega)|^p (1+ |\omega|^p)^{\gamma} d\omega<\infty \Biggr\}. \] A new technique for estimating \(s_p\) is proposed which is independent of whether the \(h_n\) are finite in number or not and it generalizes easily to the multidimensional case. The paper proposes three different methods to compute the spectral radius of the transfer operator which is known to be closely related to the Sobolev regularity index \(s_p\). In the third procedure, \(s_p\) is found by determining the zero with the smallest absolute value of the corresponding Fredholm determinant. The proposed technique can be extended to a direct computation of the Hölder exponent \(\mu\) of \(\varphi\).