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A scale of almost periodic functions spaces. (English) Zbl 1240.42115

Let \(AP_1(\mathbb R,\mathbb C)\) be the space of almost periodic functions (a.p.f.) \(f\) which are representable in the form \(f(t)=\sum _{k=1}^{\infty }A_{k}e^{i\lambda _{k} t}\), where \(A_{k} \in \mathbb C\), \(\lambda _{k} \in \mathbb {R}\) and \(\sum _{k=1}^{\infty }| A_k| < \infty \). By C. Corduneanu [Almost Periodic Oscillations and Waves, Springer (2009; Zbl 1163.34002)] \(AP_1(\mathbb R,\mathbb C)\) is a normed Banach algebra. Let \(AP_2(\mathbb R,\mathbb C)\) be the space of Besicovitch’s a.p.f. of order 2. For \(1<r<2\) the author denotes by \(AP_r(\mathbb R,\mathbb C)\) the set of a.p.f. \(f\) in \(AP_2(\mathbb R,\mathbb C)\) such that \(f(t) \sim \sum _{k=1}^{\infty }A_ke^{i \lambda _{k} t}\), where \(\sum _{k=1}^{\infty }| A_k| ^r < \infty \). Then \(AP_1(\mathbb R,\mathbb C) \subset AP_r(\mathbb R,\mathbb C) \subset AP_s(\mathbb R,\mathbb C) \subset AP_2(\mathbb R,\mathbb C)\) for \(1<r<s<2\) and \(AP_r(\mathbb R,\mathbb C)\) is obtained by completing the space of trigonometric polynomials normed with Minkowski’s norm. Besides \(AP_r(\mathbb R,\mathbb C)\) has the Bochner property and the Bohr property, and if \(f \in AP_r(\mathbb R,\mathbb C)\) is such that \(f(t) \sim \sum _{k=1}^{\infty }A_ke^{i \lambda _k t}\) and \(| \lambda _k| \geq m >0\) for \(k=1,2,\cdots \), then \(f\) has an integral in \(AP_r(\mathbb R,\mathbb C)\). The author next gives conditions on a constant matrix \(A\) of order \(m \times n\) and an operator \(F\) acting on \(AP_r(\mathbb R,\mathbb C)\) which guarantee that the systems \(x'(t)=Ax(t)+ f(t)\) (\(f \in AP_{r}(\mathbb R,\mathbb C)\)) and \(x'(t)=A x(t) + (Fx)(t)\) have a unique solution in \(AP_r(\mathbb R,\mathbb C)\). At the end of the paper, open problems are listed.

MSC:

42B35 Function spaces arising in harmonic analysis
42A75 Classical almost periodic functions, mean periodic functions
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations

Citations:

Zbl 1163.34002
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