ON THE FAILURE OF WALSH’S EQUICONVERGENCE THEOREM FOR JORDAN DOMAINS
Abstract: Let Ε be a compact set. of the complex plane with a complement E which is simply connected in the extended complex plane so that there ex is t s a conformal map ψ of the exterior of the closed unit disk D onto E c with φ(β>) = oo and ψ'(οο) > 0. Furthermore, we require that ψ admits a continuous extension onto D. Now let CR = {v(w): Iwl = K) (R > 1) be a level curve, and f be a function holomorphic in GH = Int, CR. If Ln denotes the Lagrange polynomial interpolating f in the ( n + l ) t h Fejer nodes on E, and Sn denotes the n t h partial sum of the Faber expansion of f with respect to E, then it is quite natural to ask whether there exis ts a number λ > R such that the difference Ln(z) Sn(z) tends to zero as η -»oo for all ζ e G v because the well-known equiconvergence theorem of Walsh [6, p. 153] s tates that this is valid in the case Ε = Β with λ = R*\ In this paper we will show that in a certain class of compact se t s Ε such an equiconvergence theorem only holds if Ε = B.