MERGELYAN TYPE THEOREMS FOR SOME FUNCTION SPACES

Let F be a relatively closed subset of the unit disc D. If A is any of the Hardy spaces Hp(D), 0 < p < ∞, A|F denotes the functions on F being uniform limits of elements from Hp(D). Let F ̃ consist of all z ∈ D such that |f(z) | ≤ sup{|f(z)|z ∈ F} for any bounded analytic function in D. It is proved that A|F consist of all functions f that can be decomposed as f = u + v, where u belongs to Hp(D) and v is a uniformly continuous function on the set F ̃ , analytic at interior points of F ̃. Let A be a linear space of analytic functions and F a subset of the complex plane C such that each f ∈ A is defined on F. We denote by A|F the functions being uniformly approximable on F by sequences from A. The aim with this paper is to give a partial solution to problem 8.5 no. 2 in [7]. If A is any of the classical Hardy spaces Hp(D), 0 < p <∞, our main result is that A|F coincides (modulo the approximating space) with a well defined algebra of uniformly continuous analytic functions on F. Before giving a precise formulation of the main result, we need some definitions. Let Cua(F) denote the functions on F being analytic in its interior F 0 and admitting continuous extension to the extended complex plane C∪{∞}. If F is a compact subset of C and P consists of the polynomials, a famous theorem of S. N. Mergelyan [7] can be formulated as P |F = Cua(F ̃) where F ̃ is the union of F and the bounded components of C/F. 62 A. Stray Suppose now we replace P by the set H(C) consisting of all entire functions. Also allow F to be a closed but possibly unbounded subset of C. Then it can be proved that (I): H(C)|F = H(C) + Cua(F ̃) where F ̃ is the union of F and certain components of C/F. A component V is to be included in F ̃ if and only if V ∪{∞} is not arcwise connected in C ∪ {∞}. For details see [8], [9] and [10]. In general A may contain unbounded functions. For this reason it is natural to look for an identity like (I) if we seek to describe A|F in terms of uniformly continuous analytic functions. Let us use the notation ‖g‖B = sup{|g(x) | : z ∈ B} if g is a function defined on the set B. We also define the hull of F with respect to A: F̂A = {z: |f(z) | ≤ ‖f‖F, f ∈ A}. We look for spaces A satisfying the following