Pseudohyperbolic distance and n-best rational approximation in H^2 space

Through reducing the problem to rational orthogonal system (Takenaka-Malmquist system), this note gives a proof for existence of n-best rational approximation to functions in the Hardy H^2(D) space by using pseudohyperbolic distance.

REMARKS ON ADAPTIVE FOURIER DECOMPOSITION

This is a continuation of the study of adaptive Fourier decomposition (AFD).15 Under a mild condition not in terms of smoothness, a convergence rate is provided. We prove that the selection of the parameters corresponding to Fourier series in the average sense is optimal. We also present the transformation matrices between the adaptive rational orthogonal system and the related sequence of the shifted Cauchy kernels and their derivatives.