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.


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


Zbl 1163.34002