Approximation and interpolation

Abstract

Let X be a compact plane set, X° its interior, and suppose E is a subset of dX = X\X°. H°°(X 0 ) is the algebra of all bounded analytic functions on X° and H£(X°) denotes all bounded continuous functions on X° u E analytic in X°.Interpolation sets for HE(X°) are studied if E is open relative to dX.If X satisfies certain conditions which involve analytic capacity, it is shown that an interpolation set S for H°°(X°) is an interpolation set for ϋ°°(0) for some open set 0 which contains every point of X except the points on dX in the closure of S. Similar results are proved for R(X) without restrictions on X.Introduction and notation* The paper is divided into three

Approximation and interpolation for some spaces of analytic functions in the unit disc

“…Fix an integer N and choose/ G H P (T). As in [4] page 204 and 139-140, we get that the function g N in H P (D) of minimal norm which interpolates/on {z u ..., z N } must satisfy (11) H^ll, < IW …”

“…Indeed if Lemma 1 is proved, Theorem 2 follows by the same iteration argument as in the proof of Lemma 1.4 in [11].…”

“…The first part of the proof of Lemma 1 is very similar to the proof of Lemma 3.2 in [11], but for completeness we give most of the details.…”

“…If one applies Lemma 2 with care and approximate the functions R kt \ and R k 2 by functions S kt x and S kt 2 analytic in C\(S\D) using that lemma, we get "new" functions /"** and g n ** by replacing R kf x and R k2 by S kt { and S kj 2 in the expressions of the form (4) for/,* and g n *. Define then (11) :Λi = Λ Note from property (v) of { V n } that if UD (S\D) is open then there exists a number N such that the poles of the rational functions R k> x and R kt 2 corresponding tof n * and g n *, must be contained in U if n > N. From this fact and (ii) in Lemma 2 it is easy to see that the series (11) will converge uniformly on compact subsets of C\(S\D). From (9) and (10) it follows that hi will satisfy Lemma 1 if Lemma 2 is applied carefully.…”

Interpolation sets for analytic functions

“…We observe that if A(U) is pointwise boundedly dense in iϋΓ°°(U) then using Theorem 2.1 of [5] we can modify the function / in the lemma so that it is in Hp um \f } (U).…”