The Caratheodory--Fejer Extension of a Finite Geometric Series

It is shown that the Caratheodory—Fejer extension of afinite geometric series can be given explicitly up to a simplepolynomial equation in an auxiliary variable. This result allowsus to analyse the Caratheodory-Fejer approximation method inthe case where the quotients of successive Maclaurin coefficientsof the given function tend to a limit. ^*Research carried out while this author was at ETH Zurich partiallysupported by a Royal Society European Visiting Fellowship.