## 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

Zbl 1163.34002