A de Montessus-type theorem for CF approximation

摘要

The following de Montessus-type theorem for Carathéodory—Fejér (CF) approximants is proven: Let f be meromorphic in ∣ z ∣ < ρ (ρ>1), analytic in ∣z∣ ⩽ 1, and with a total of n poles ζ1,…,ζn (multiplicity included) in 1 < ∣ z ∣ < ρ. Then, as m → ∞, the CF approximants rcfmn of f from Rmn converge on {z∈C;|z| {ρ}⧹{ζ1,…ζn} to ƒ uniformly on every compact subset. Here, rcfmn may be either the type 1 or the type 2 CF approximant, and a similar result holds for the untruncated CF approximant.