Rational approximation in Orlicz spaces on Carleson curves

Abstract

AbstractWe define some subclasses of Orlicz spaces of functions and establish herea direct theorem of the approximation theory by rational functions. 1 Introduction and main results Let Γ be a rectifiable Jordan curve in the complex plane C and let G:= IntΓ,G − := ExtΓ. Without loss of generality we suppose that 0 ∈ G.Further let T :={w∈ C : |w| = 1}, U:= IntT,U − := ExtT.We denote by ϕand ϕ 1 the conformalmappings of G − and Gonto U − normalized by the conditionsϕ(∞) = ∞, lim z→∞ ϕ(z)z>0andϕ 1 (0) = ∞, lim z→0 zϕ 1 (z) >0respectively and let ψand ψ 1 be the inverse mappings of ϕand ϕ 1 .Let also L p (Γ) and E p (G) (1 ≤ p<∞) be the Lebesgue space of measurablecomplex valued functions on Γ and the Smirnov class of analytic functions in Grespectively. Since Γ is rectifiable, we have ϕ 0 ∈ E 1 (G − ), ϕ 0 1 ∈ E 1 (G) and ψ 0 , Received by the editors June 2003.Communicated by R. Delanghe.1991 Mathematics Subject Classification : Primary 30E10, 41A10, 41A20. Secondary 41A25,46E30.Key words and phrases : Carleson curve, Cauchy singular operator, Faber polynomials, Orliczspace, Smirnov-Orlicz class.Bull. Belg. Math. Soc. 12(2005), 223–234