Uniform subalgebras of $L^{\infty }$ on the unit circle generated by almost periodic functions

Abstract

Abstract. Analogs of almost periodic functions for the unit circle are introduced. Certain uniform algebras generated by such functions are studied, the corona theorems for them are proved, and their maximal ideal spaces are described. §1. Formulation of main results 1.1.The classical almost periodic functions on the real line, as first introduced by H. Bohr in the 1920s, play an important role in various areas of analysis. In the present paper we define analogs of almost periodic functions on the unit circle. We study certain uniform algebras generated by such functions. In particular, in these terms we describe some uniform subalgebras of the algebra H ∞ of bounded holomorphic functions on the open unit disk D ⊂ C that, in a sense, have the weakest possible discontinuities on the boundary ∂D.To formulate the main results of the paper, we start with recalling the definition of an almost periodic function; see [B]. Definition 1.1. A continuous function f : R → C is said to be almost periodic if, for any > 0, there exists l( ) > 0 such that for every t 0 ∈ R the interval [t 0 , t 0 + l( )] contains at least one number τ for whichIt is well known that every almost periodic function f is uniformly continuous and is the uniform limit of a sequence of exponential polynomials {q n } n∈N , where q n (t) := n k=1 c kn e iλ kn t , c kn ∈ C, λ kn ∈ R, 1 ≤ k ≤ n, and i := √ −1. In what follows we consider ∂D with the counterclockwise orientation.as the right or the left endpoint relative to the chosen orientation, respectively. We define almost periodic functions on open arcs of ∂D. 2000 Mathematics Subject Classification. Primary 30H05; Secondary 46J20.

Holomorphic Semi-almost Periodic Functions

“…Finally, we prove that SAP (∂D) ∩ H ∞ (D) has the approximation property. (This result strengthen the approximation theorem of [6].) Recall that a Banach space B is said to have the approximation property if for every compact set K ⊂ B and every ε > 0 there is an operator T : B → B of finite rank so that T x − x B < ε for every x ∈ K.…”

“…The proof of the next statement can be obtained by following closely the arguments in the proof of Theorem 1.14 in [6]. In its formulation we assume that the corresponding algebras are defined on their maximal ideal spaces by means of the Gelfand transforms.…”

“…Proof of Theorem 3.11. Corollary 1.6 in [6] states that κ * | ∂D : C(∂D) → C(∂D), the pullback by κ| ∂D , maps SAP (κ * S) isomorphically onto SAP (S). Following closely the arguments in its proof, one obtains even more: κ * maps SAP κ * Σ (κ * S) isomorphically onto SAP Σ (S).…”

“…Our proof of the approximation property for SAP (S) ∩ H ∞ (D) is based on Theorem 1.1 and on a variant of the approximation theorem in [6] for Banach-valued analogues of algebra SAP (S) ∩ H ∞ (D).…”

Towards Oka–Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds I

“…Using a normal family argument one shows thatf ∈ O(c a X), see, e.g., [Lin] or [BrK2,Lemma 2.3] for similar results. Clearly, ι * f = f .…”