Ukrainian Mathematical Journal, Vol. 56, No. 7, 2004 BEST APPROXIMATION OF REPRODUCING KERNELS OF SPACES OF ANALYTIC FUNCTIONS V. V. Savchuk UDC 517.5 We obtain exact values for the best approximation of a reproducing kernel of a system of p- Faber polynomials by functions of the Hardy space H q , p q − − + 1 1 = 1, and a Szegö reproduc- ing kernel of the space E 2 ( ) Ω in a simply connected domain Ω with rectifiable boundary. 1. Introduction Let Γ be a closed rectifiable Jordan curve in the complex plane C. By Ω and Ω ∞ we denote the compo- nents int Γ and ext Γ of the doubly-connected domain ˆ \ C Γ on the Riemann sphere ˆ C : = C ∪ {∞ } ; the first of these components (the one that does not contain an infinitely remote point ∞) is called the interior of the curve Γ, and the second component is called the exterior of the curve Γ. In the case where Γ coincides with the unit circle, i.e., Γ = T : = {z ∈ C : | z | = 1 }, we write D and D ∞ instead of Ω and Ω ∞ , respectively. By virtue of the Riemann theorem, there exists a unique function Ψ holomorphic in D ∞ that conformally maps the domain D ∞ onto the domain Ω ∞ , provided that Ψ ( ∞ ) = ∞ and lim ( ) w w w →∞ −1 Ψ = γ > 0. We set K p ( w, z ) : = w w w z p [ ] ( ) ( ) / ′ − − Ψ Ψ 11 . If 1 ≤ p ≤ ∞ and z ∈ Ω, then a function w  K p ( w, z ) , w ∈ D ∞ , is called a reproducing kernel of a system of (generalized) p-Faber polynomials { } , F kpk = ∞ 0 corresponding to the domain Ω (see, e.g., [1, pp. 369, 375]), and, furthermore, K p ( w, z ) = F zw kp k k , () − = ∞ ∑ 0 ∀ w ∈ D ∞ , (1) where series (1) converges uniformly in the domain D ∞ for every fixed z in Ω. Let 0 < p < ∞ , let σ be a normalized Lebesgue measure on the circle T, and let L p ( T ) be the space of functions f : T → C measurable with respect to σ and satisfying the condition f L p ( ) T : = fw d w p p ( ) ( ) / σ T ∫       1 < ∞. Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 7, pp. 947–959, July, 2004. Original article submitted June 3, 2003. 0041–5995/04/5607–1127 © 2004 Springer Science+Business Media, Inc. 1127