In this chapter we give some applications of the results from Chapters 1–5. These amply illustrate that Reg is an important and natural class of measures that appears in different problems of mathematical analysis. In Section 6.1 we consider rational interpolants to Markov functions and see how regularity of the measure defining the Markov function in question is related to the rate of convergence of the interpolants. Section 6.2 characterizes the regularity of a measure generating a Markov function by an exact, maximal rate of convergence of best rational approximants on compact sets. In Section 6.3 we consider similar questions but for ray sequences of Padé approximants. An interesting feature of the proof is that the problem on the upper half of the Padé table is reduced to that on the lower half by the localization theorems of the preceding chapter. In Sections 6.4 and 6.5 we connect regularity to weighted polynomials and answer the question of where the Lp norm of these weighted polynomials lives. Section 6.6 is devoted to the relation of μ ∈ Reg to Fourier coefficients and best L2 polynomial approximation of analytic functions. Finally, in Section 6.7 we investigate sets E that have the property that orthonormal polynomials with respect to any weight that is positive on E have regular nth-root behavior.
Rational Interpolants to Markov Functions
Orthogonal polynomials are closely related to continued fractions, and one of the classical results in the analytic theory of continued fractions is Markov's theorem (see [Ma] or [Pe]).