Fast Analytic Sampling Approximation from Cauchy Kernel

Abstract

The paper aims at establishing a fast numerical algorithm for Bk(f), where f is any function in the Hardy space H2(Td) and k is the scale level. Here, Bk(f) is an approximation to f we recently constructed by applying the multiscale transform to the Cauchy kernel. We establish the matrix expression of Bk(f) and find that it has the structure of a multilevel Hankel matrix. Based on the structure, a fast numerical algorithm is established to compute Bk(f). The computational complexity is given. A numerical experiment is carried out to check the efficiency of our algorithm.