Bohr’s equivalence relation in the space of Besicovitch almost periodic functions. (English) Zbl 1420.42004
Summary: Based on Bohr’s equivalence relation which was established for general Dirichlet series, in this paper we introduce a new equivalence relation on the space of almost periodic functions in the sense of Besicovitch, \(B(\mathbb{R},\mathbb{C})\), defined in terms of polynomial approximations. From this, we show that in an important subspace \(B^2(\mathbb{R},\mathbb{C})\subset B(\mathbb{R},\mathbb{C})\), where Parseval’s equality and the Riesz-Fischer theorem hold, its equivalence classes are sequentially compact and the family of translates of a function belonging to this subspace is dense in its own class.
MSC:
| 42A75 | Classical almost periodic functions, mean periodic functions |
| 42A16 | Fourier coefficients, Fourier series of functions with special properties, special Fourier series |
| 42B05 | Fourier series and coefficients in several variables |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 30B50 | Dirichlet series, exponential series and other series in one complex variable |
Keywords:
almost periodic functions; Besicovitch almost periodic functions; Bochner’s theorem; exponential sums; Fourier seriesReferences:
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