Besicovitch-Orlicz spaces of almost periodic functions.

*(English)*Zbl 0656.46020
Real and stochastic analysis, Wiley Ser. Probab. Math. Stat., Probab. Math. Stat., 119-167 (1986).

[For the entire collection see Zbl 0588.00021.]

H. Bohr’s almost periodic functions are the elements of the closure of trigonometric polynomials in the set of continuous functions on the real line under the uniform distance. The generalization \(S^ p\) a.p., \(W^ p\) a.p., and \(B^ p\) a.p. of Stepanoff, Weyl, and Besicovitch, respectively, are the closures of trigonometric polynomials in the Lebesgue space \(L^ p_{loc}({\mathbb{R}})\) (p\(\geq 1)\) under distances related to the \(L^ p\)-norm.

A completion of the theory concerning these generalized almost periodic functions is obtained in this work by considering what are called \(S^{\Phi}\) a.p. (Stepanoff-Orlicz almost periodic), \(W^{\Phi}\) a.p. (Weyl-Orlicz almost periodic), and \(B^{\Phi}\) a.p. (Besicovitch-Orlicz almost periodic) functions. These spaces are the closures of trigonometric polynomials in the Orlicz space \(L^{\Phi}_{loc}({\mathbb{R}})\) under distances related to the Orlicz norm. Structural characterizations and approximation theorems of the new spaces of a.p. functions are given. A brief outline of the results is as follows.

Section 1 deals with the preliminary notions and the definitions of these spaces. It is shown that these functions are approximable by Bochner-Fejér polynomials. Section 2 contains the structure theorems. Section 3 contains the completeness proofs of the \(S^{\Phi}\) a.p. and \(B^{\Phi}\) a.p. and the incompleteness of the \(W^{\Phi}\) a.p. Section 4 deals with the correspondence theorems and duality theory of \(B^{\Phi}\) a.p. spaces.

H. Bohr’s almost periodic functions are the elements of the closure of trigonometric polynomials in the set of continuous functions on the real line under the uniform distance. The generalization \(S^ p\) a.p., \(W^ p\) a.p., and \(B^ p\) a.p. of Stepanoff, Weyl, and Besicovitch, respectively, are the closures of trigonometric polynomials in the Lebesgue space \(L^ p_{loc}({\mathbb{R}})\) (p\(\geq 1)\) under distances related to the \(L^ p\)-norm.

A completion of the theory concerning these generalized almost periodic functions is obtained in this work by considering what are called \(S^{\Phi}\) a.p. (Stepanoff-Orlicz almost periodic), \(W^{\Phi}\) a.p. (Weyl-Orlicz almost periodic), and \(B^{\Phi}\) a.p. (Besicovitch-Orlicz almost periodic) functions. These spaces are the closures of trigonometric polynomials in the Orlicz space \(L^{\Phi}_{loc}({\mathbb{R}})\) under distances related to the Orlicz norm. Structural characterizations and approximation theorems of the new spaces of a.p. functions are given. A brief outline of the results is as follows.

Section 1 deals with the preliminary notions and the definitions of these spaces. It is shown that these functions are approximable by Bochner-Fejér polynomials. Section 2 contains the structure theorems. Section 3 contains the completeness proofs of the \(S^{\Phi}\) a.p. and \(B^{\Phi}\) a.p. and the incompleteness of the \(W^{\Phi}\) a.p. Section 4 deals with the correspondence theorems and duality theory of \(B^{\Phi}\) a.p. spaces.

Reviewer: V.Zakharov

##### MSC:

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |