Interpolation properties of rational functions of best mean square approximation on a circle and in a disk
Consider the usual Hilbert space \(L_2(T)\) of square integrable functions with respect to the Lebesgue measure on the unit circle \(T\). The Cauchy transform of \(f\in L_2(T)\) defines an analytic function \(F\) in the unit disk and \(G\) in the complement of the unit disk. Mean square best rational approximation of \(f\) is considered of all possible degrees in the numerator and denominator. Such best approximants \(r\) are decomposed in simple fractions and separated the part with poles in the unit disk \(\tau\) from the part with poles on the exterior of the unit disk \(\lambda\) so that \(\tau + \lambda=r\). Exact results are provided on the extent to which \(\tau\) interpolates \(G\) and \(\lambda\) interpolates \(F\).
scientific article; zbMATH DE number 846478