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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.
Reviewer: V.Zakharov
MSC:
 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems