Fast Analytic Sampling Approximation from Cauchy Kernel

Abstract

The paper aims at establishing a fast numerical algorithm forBk(f), wherefis any function in the Hardy spaceH2(Td)andkis the scale level. Here,Bk(f)is an approximation tofwe recently constructed by applying the multiscale transform to the Cauchy kernel. We establish the matrix expression ofBk(f)and find that it has the structure of a multilevel Hankel matrix. Based on the structure, a fast numerical algorithm is established to computeBk(f). The computational complexity is given. A numerical experiment is carried out to check the efficiency of our algorithm.