In this treatise, the authors present the general theory of orthogonal polynomials on the complex plane and several of its applications. The assumptions on the measure of orthogonality are general, the only restriction is that it has compact support on the complex plane. In the development of the theory the main emphasis is on asymptotic behaviour and the distribution of zeros. In the following chapters, the author explores the exact upper and lower bounds are given for the orthonormal polynomials and for the location of their zeros; regular n-th root asymptotic behaviour; and applications of the theory, including exact rates for convergence of rational interpolants, best rational approximants and non-diagonal Pade approximants to Markov functions (Cauchy transforms of measures). The results are based on potential theoretic methods, so both the methods and the results can be extended to extremal polynomials in norms other than L2 norms. A sketch of the theory of logarithmic potentials is given in an appendix.

Pade approximants are a natural generalization of Taylor polynomials; however instead of polynomials now rational functions are used for the development of a given function. In this article the convergence in capacity of Pade approximants [mÂn] with m+n Ä , mÂn Ä 1, is investigated. Two types of assumptions are considered: In the first case the function f to be approximated has to have all its singularities in a compact set E C of capacity zero (the function may be multi&valued in C "E). In the second case the function f has to be analytic in a domain possessing a certain symmetry property (this notion is defined and discussed below). It is shown that close-to-diagonal sequences of Pade approximants [mÂn] converge to f in capacity in a domain D that can be determined in various ways. In the case of the first type of assumptions the domain D is determined by the minimality of the capacity of the complement of D, in the second case the domain D is determined by a symmetry property. The rate of convergence is determined, and it is shown that this rate is best possible for convergence in capacity. In addition to the convergence results the asymptotic distribution of zeros and poles of the approximants is studied.1997 Academic Press

In this paper we continue our study of the asymptotic behavior of polynomials Qm.(z), m, n c N, of degree -~ n satisfying the orthogonal relation (*) ~c~lQ,.,,(~) ~=O, l=O,.,.,n-1, where the function f(z) is supposed to be analytic on a continuum V~ C and all its singularities are supposed to be contained in a set E ~ C of capacity zero, to,,+,(z) is a polynomial of degree m + n + 1 with all its zeros contained in V, and the path of integration C separates V from the set E. We state and prove results concerning the asymptotic magnitude of the integral in (*) for l= n,n+l, .... Main Results In [ 1 ]we investigated the asymptotic distribution of the zeros of nontrivial polynomials Q.,n of degree < n which satisfy (1.1) ~c~lQmn(~) f(~! d~=O, l=O,...,n-1, where m, n ~ N, and the function f(z) is supposed to be analytic on a simply connected continuum V~ (2 and to have all its singularities in a set E c_ ~ of (logarithmic) capacity zero. Further it is supposed that among the singularities there are branch points. The path of integration C separates the continuum V from E, and tom+, is a given polynomial of degree m + n + 1 with all its zeros contained in V.It has been shown [1, Theorem 1] that if the m + n + 1 zeros of o~m+~ possess a certain asymptotic distribution for m + n -> oo and m/n --> 1, then the asymptotic distribution of the zeros of the polynomials Qm, is essentially determined by relation (1.

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