Weighted polynomials and weighted pluripotential theory

Bloom, Thomas

doi:10.1090/S0002-9947-08-04607-2

Weighted polynomials and weighted pluripotential theory
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Trans. Amer. Math. Soc. 361 (2009), 2163-2179 Request permission

Abstract:

Let $E$ be a compact subset of $\mathbb {C} ^{N}$ and $w\ge 0$ an admissible weight function on $E$. To $(E, w)$ we associate a canonical circular set $Z\subset \mathbb {C} ^{N+1}$. We obtain precise relations between the weighted pluricomplex Green function and weighted equilibrium measure of $(E, w)$ and the pluricomplex Green function and equilibrium measure of $Z$. These results, combined with an appropriate form of the Bernstein-Markov inequality, are used to obtain asymptotic formulas for the leading coefficients of orthonormal polynomials with respect to certain exponentially decreasing weights in $\mathbb {R}^{N}$.

References

H. Alexander, Projective capacity, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981, pp. 3–27. MR 627747
Thomas Bloom, Orthogonal polynomials in $\mathbf C^n$, Indiana Univ. Math. J. 46 (1997), no. 2, 427–452. MR 1481598, DOI 10.1512/iumj.1997.46.1360
Thomas Bloom, Random polynomials and Green functions, Int. Math. Res. Not. 28 (2005), 1689–1708. MR 2172337, DOI 10.1155/IMRN.2005.1689
T. Bloom and N. Levenberg, Capacity convergence results and applications to a Bernstein-Markov inequality, Trans. Amer. Math. Soc. 351 (1999), no. 12, 4753–4767. MR 1695017, DOI 10.1090/S0002-9947-99-02556-8
T. Bloom and N. Levenberg, Weighted pluripotential theory in $\mathbf C^N$, Amer. J. Math. 125 (2003), no. 1, 57–103. MR 1953518
T. Bloom, N. Levenberg, and S. Ma’u, Robin functions and extremal functions, Proceedings of Conference on Complex Analysis (Bielsko-Biała, 2001), 2003, pp. 55–84. MR 1972834, DOI 10.4064/ap80-0-4
P. A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, vol. 3, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. MR 1677884
Laura DeMarco, Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity, Math. Ann. 326 (2003), no. 1, 43–73. MR 1981611, DOI 10.1007/s00208-002-0404-7
Mieczysław Jędrzejowski, The homogeneous transfinite diameter of a compact subset of $\textbf {C}^N$, Proceedings of the Tenth Conference on Analytic Functions (Szczyrk, 1990), 1991, pp. 191–205. MR 1141434, DOI 10.4064/ap-55-1-191-205
Maciej Klimek, Pluripotential theory, London Mathematical Society Monographs. New Series, vol. 6, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR 1150978
Józef Siciak, Extremal plurisubharmonic functions in $\textbf {C}^{n}$, Ann. Polon. Math. 39 (1981), 175–211. MR 617459, DOI 10.4064/ap-39-1-175-211
Józef Siciak, On series of homogeneous polynomials and their partial sums, Ann. Polon. Math. 51 (1990), 289–302. MR 1094001, DOI 10.4064/ap-51-1-289-302
Józef Siciak, A remark on Tchebysheff polynomials in $\mathbf C^N$, Univ. Iagel. Acta Math. 35 (1997), 37–45. MR 1458043
Edward B. Saff and Vilmos Totik, Logarithmic potentials with external fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom. MR 1485778, DOI 10.1007/978-3-662-03329-6
Herbert Stahl and Vilmos Totik, General orthogonal polynomials, Encyclopedia of Mathematics and its Applications, vol. 43, Cambridge University Press, Cambridge, 1992. MR 1163828, DOI 10.1017/CBO9780511759420
Thanh Van Nguyen and Ahmed Zériahi, Familles de polynômes presque partout bornées, Bull. Sci. Math. (2) 107 (1983), no. 1, 81–91 (French, with English summary). MR 699992