General Orthogonal Polynomials (Encyclopedia of Mathematics and its Applications, Series Number 43)

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. Herbert Stahl, Vilmos Totik. Includes Bibliographical References And Index. Title page Preface Acknowledgments Symbols 1 Upper and Lower Bounds 1.1 Statement of the Main Results 1.2 Some Potential-theoretic Preliminaries 1.3 Proof of the Upper and Lower Bounds 1.4 Proof of the Sharpness of the Upper and Lower Bounds 1.5 Examples 2 Zero Distribution of Orthogonal Polynomials 2.1 Zeros of Orthogonal Polynomials 2.2 Norm Asymptotics and Zero Distribution 2.3 Asymptotic Behavior of Zeros when c_μ>0 3 Regular nth-root Asymptotic Behavior of Orthonormal Polynomials 3.1 Regular Asymptotic Behavior 3.2 Characterization of Regular Asymptotic Behavior 3.3 Regular Behavior in the Case of Varying Weights 3.4 Characterization of Regular Asymptotic Behavior in L^p(μ) 3.5 Examples 3.6 Regular Behavior and Monic Polynomials 4 Regularity Criteria 4.1 Existing Regularity Criteria and Their Generalizations 4.2 New Criteria and Their Sharpness 4.3 Proof of the Regularity Criteria 4.4 Preliminaries for Proving the Sharpness of the Criteria 4.5 Proof of the Sharpness of the Regularity Criteria 4.6 Summary of Regularity Criteria and Their Relations 5 Localization 5.1 Global versus Local Behavior 5.2 Localization at a Single Point 5.3 Localization Theorems 6 Applications 6.1 Rational Interpolants to Markov Functions 6.2 Best Rational Approximants to Markov Functions 6.3 Nondiagonal Padé Approximants to Markov Functions 6.4 Weighted Polynomials in L^p(μ) 6.5 Regularity and Weighted Chebyshev Constants 6.6 Regularity and Best L2(μ) Polynomial Approximation 6.7 Determining Sets Appendix A.I Energy and Capacity A.II Potentials, Fine Topology A.III Principles A.IV Equilibrium Measures A.V Green Functions A.VI Dirichlet's Problem A.VII Balayage A.VIII Green Potential and Condenser Capacity A.IX The Energy Problem in the Presence of an External Field Notes and Bibliographical References Bibliography Index 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 behavior and the distribution of zeros. In the first two chapters exact upper and lower bounds are given for the orthonormal polynomials and for the location of their zeros. The next three chapters deal with regular n-th root asymptotic behavior, which plays a key role both in the theory and in its applications. Orthogonal polynomials with this behavior correspond to classical orthogonal polynomials in the general case, and many extremal properties of measures in mathematical analysis and approximation theory with this type of regularity turn out to be equivalent. Several easy-to-use criteria are presented for regular behavior. The last chapter contains 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 L]2 norms. A sketch of the theory of logarithmic potentials is given in an appendix. An encyclopedic presentation of general orthogonal polynomials, placing emphasis on asymptotic behaviour and zero distribution