Abstract:
This paper investigates rational approximations of a Markov function that have the highest order of contact with it at infinity, and whose denominators are invariant under multiplication of their argument by a root of unity of some fixed degree (such approximations are used in many number-theoretical problems). The approximations converge under mild restrictions on the measure. Moreover, the denominators of the approximants and the corresponding functions of the second kind have logarithmic asymptotics expressible in terms of a certain extremal measure which, in the simplest case, is the Tchebycheff measure. An explicit form is found for the extremal measure in the general case; in fact, the inverse of the distribution function is expressed in terms of elementary functions, the power moments are calculated, and the Markov function of the extremal measure is connected with algebraic equations and generalized hypergeometric functions.
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