1 - Upper and Lower Bounds

Summary

This chapter is devoted to sharp upper and lower bounds for orthonormal polynomials with respect to general weights. The two bounds are given in terms of Green functions related to the carriers of the measure in question. As a corollary, sharp bounds are obtained for the leading coefficients. All subsequent chapters use both the notations and the results from the present one.

The chapter is organized as follows: Section 1.1 contains the statement of the main results; in Section 1.2 we prove some potential-theoretic preliminaries needed in the proofs. The actual proofs of the upper and lower estimates are carried out in Section 1.3, and the proof of their sharpness is given in Section 1.4. Finally, in Section 1.5 we construct some examples that illustrate the results.

Statement of the Main Results

The main results in this section are lower and upper asymptotic bounds for the nth root of the orthonormal polynomials p_n(μ; z) as n → ∞, as well as their unimprovability.

In what follows cap(S) denotes the (outer logarithmic) capacity of a bounded set S ⊆ ℂ, that is, cap(S) = inf_U cap(U), where the infimum extends over all open sets U ⊇ S (see Chapter 11, Section 2 of [La] or Appendix I), and we say that a property holds qu.e. (quasi everywhere) on a set S ⊆ ℂ if it holds on S with possible exceptions on a subset of capacity zero.