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.
General
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