Uniform Subalgebras of L∞ on the Unit Circle Generated by Almost Periodic Functions

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 t0 ∈ R the interval [t0, t0 + l( )] contains at least one number τ for which |f(t)− f(t+ τ )| < for all t ∈ R. It is well known that every almost periodic function f is uniformly continuous and is the uniform limit of a sequence of exponential polynomials {qn}n∈N, where qn(t) := ∑n k=1 ckne kn, ckn ∈ C, λkn ∈ R, 1 ≤ k ≤ n, and i := √ −1. In what follows we consider ∂D with the counterclockwise orientation. For t0 ∈ R, let γtk0 (s) := {e i(t0+kt) : 0 < t < s ≤ 2π} ⊂ ∂D, k ∈ {−1, 1}, be two open arcs having e0 as the right or the left endpoint relative to the chosen orientation, respectively. We define almost periodic functions on open arcs of ∂D. Definition 1.2. A continuous function fk : γtk0 (s) → C, k ∈ {−1, 1}, is said to be almost periodic if the function f̂k : (−∞, 0) → C, f̂k(t) := fk(e0 t)), is the restriction of an almost periodic function on R. Example 1.3. In the sense of this definition, the function e iλ log t0 , λ ∈ R, where logtk0 (e 0) := ln t, 0 < t < 2π, k ∈ {−1, 1}, is almost periodic on γt10(2π) = γt−1 0 (2π). 2000 Mathematics Subject Classification. Primary 30H05; Secondary 46J20.