Siberian Mathematical Journal, Vol. 46, No. 4, pp. 613–622, 2005 Original Russian Text Copyright c  2005 Dovgoshe˘ ı A. A., Abdullaev F., and K¨ u¸ c¨ ukaslan M. THE LOGARITHMIC ASYMPTOTIC EXPANSIONS FOR THE NORMS OF EVALUATION FUNCTIONALS A. A. Dovgoshe˘ ı, F. Abdullaev, and M. K¨ u¸ c¨ ukaslan UDC 517.538.3 Abstract: Let μ be a compactly supported finite Borel measure in C, and let Π n be the space of holo- morphic polynomials of degree at most n furnished with the norm of L 2 (μ). We study the logarithmic asymptotic expansions of the norms of the evaluation functionals that relate to polynomials p ∈ Π n their values at a point z ∈ C. The main results demonstrate how the asymptotic behavior depends on regularity of the complement of the support of μ and the Stahl–Totik regularity of the measure. In particular, we study the cases of pointwise and μ-a.e. convergence as n →∞. Keywords: general orthogonal polynomials, logarithmic asymptotic expansion, evaluation functionals, Green’s function, irregularity points for the Dirichlet problem 1. Introduction Suppose that μ is a finite positive Borel measure with compact support S μ ⊆ C, card S μ = ∞, and Π n =Π n (μ) is the Hilbert space of all polynomials of degree at most n in one complex variable with the norm induced from L 2 (μ). Given z 0 ∈ C, the evaluation functional F n,z 0 :Π n → C is defined by the relation F n,z 0 (p)= p(z 0 ). We denote the norm of this functional by ‖F n,z 0 ‖ μ ; i.e., ‖F n,z 0 ‖ μ = sup{|p(z 0 )| : deg p ≤ n, ‖p‖ L 2 (μ) ≤ 1}. These norms depending on n, z 0 , and μ (and not only in the case of L 2 (μ)) have been attracting the attention of many specialists in functional analysis and the theory of orthogonal polynomials. The classicalSzeg¨o–Kolmogorov–Kre˘ ın Theorem gives a condition for boundedness of the norms of F n,z 0 and completeness of the polynomials in L p (μ) for p> 0 and S μ ⊆{z : |z | =1}. Similar problems for measures with more complicated geometry of the support were studied in the articles by A. L. Vol ′ berg [1], Thomson [2], Akeroyd [3, 4], and other mathematicians. A monographic exposition of related topics can be found in [5]. Observe that the quantity 1 n+1 ‖F n,z ‖ 2 μ coincides with the arithmetic mean of the squares of the absolute values of orthonormal polynomials at z . Many interesting results on the asymptotic properties of these means in the case of measures concentrated on the real axis or the unit circle can be found in [6]. Here we study the pointwise and μ-a.e. convergences of 1 n log ‖F n,z ‖ μ as n →∞ for general compactly supported Borel measures. The logarithmic asymptotic expansions of orthogonal polynomials corresponding to such measures were studied in the fundamental monograph [7] by Stahl and Totik. In the following section of the article we formulate our results and compare them with the corre- sponding assertions of [7]. The last section contains the proofs and some lemmas. 2. The Main Results and Auxiliary Information Since μ is finite, S μ is a compact set, and card S μ = ∞; there is a unique system of polynomials p n (μ; ·) such that  p n (μ; ·) p m (μ; ·) dμ =  1 for n = m, 0 for n = m, p n (μ; z )= γ n z n + ..., Donetsk (the Ukraine); Mersin (Turkey). Translated from Sibirski˘ ı Matematicheski˘ ı Zhurnal, Vol. 46, No. 4, pp. 774–785, July–August, 2005. Original article submitted August 13, 2003. Revision submitted January 28, 2005. 0037-4466/05/4604–0613 c  2005 Springer Science+Business Media, Inc. 613