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Padé approximants, continued fractions, and orthogonal polynomials. (English. Russian original) Zbl 1242.41014
Russ. Math. Surv. 66, No. 6, 1049-1131 (2011); translation from Usp. Mat. Nauk 66, No. 6, 37-122 (2011).
This paper gives a review of the modern convergence theory of Padé approximants and related areas. The considered topics include the following:
1)
Asymptotic properties of orthogonal polynomials and the distribution of the zeros of extremal polynomials.
2)
Convergence of Padé approximants for Markov-type and hyperelliptic functions.
3)
An inverse theorem for diagonal Padé approximants.
4)
Rate of rational approximation of analytic functions.
5)
Recurrence relations, theorem of Poincaré and Perron, and their applications to the convergence theory of continued fractions.
6)
Convergence of the Rogers-Ramanujan continued fractions.
Some historical background and conjectures in the theory of Padé approximants are also given.

MSC:
41A21 Padé approximation
30B70 Continued fractions; complex-analytic aspects
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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