On Almost Periodic at Infinity Distributions from Harmonic Spaces

The article under consideration is devoted to some problems of harmonic analysis of almost periodic at infinity functions from homogeneous spaces and distributions from harmonic spaces. We introduce the definition of an abstract homogeneous space $ \mathscr{F}( \mathbb{R} ,X)$ of functions defined on real axis $\mathbb{R}$ and with values in a complex Banach space $X$.The homogeneous function spaces under investigation are contained in Stepanov space $S^{1}(\mathbb{R},X)$, convolutions with complex-valued functions from Banach algebra $L^{1}(\mathbb{R})$ and shifts do not take out any function from the corresponding homogeneous space. Thanks to the convolution with functions from $L^{1}(\mathbb{R})$ any homogeneous function space can be endowed with Banach module structure. We provide a number of examples of homogeneous function spaces including spaces $L^{p}(\mathbb{R} ,X)$, $p \in [1,\infty ]$ , Stepanov spaces $S^{1}(\mathbb{R},X)$, $p \in [1,\infty ]$, Wiener amalgam spaces $L^{p}(\mathbb{R} ,X), l^{q}(\mathbb{R} ,X), p,q \in [1,\infty )$, continuous function spaces $ C_{b}(\mathbb{R}, X), C_{b, u}(\mathbb{R}, X)$ and $C_{0}(\mathbb{R},X)$, Hoelder spaces etc. Then we introduce the notions of the subspaces $ \mathscr{F}_{sl, \infty}( \mathbb{R} ,X)$ and $AP_{\infty}\mathscr{F}( \mathbb{R} ,X)$ of slowly varying and almost periodic at infinity functions from $\mathscr{F}( \mathbb{R} ,X)$. For functions from $AP_{\infty}\mathscr{F}( \mathbb{R} ,X)$ we construct a notion of Fourier series, which is ambiguous, i.e., its coefficients can be chosen differently. We prove the Fourier coefficients to be slowly varying at infinity.

However, Borel measures on $\mathbb{R}$ with bounded variation and values in a Banach space $X$ do not satisfy the definition of a homogeneous function space and force us to introduce an applicable extension. We consider the space ${S}'(\mathbb{R},X)$ of distributions of slow growth. On the space ${S}'(\mathbb{R},X)$ we define a group of shift operators and convolution with a function from a Banach algebra $L^{1}(\mathbb{R})$. For a homogeneous space $ \mathscr{F}( \mathbb{R} ,X)$ (denoted also as $ \mathscr{F}^{0}( \mathbb{R} ,X)$ we introduce a countable set of spaces $ \mathscr{F}^{n}( \mathbb{R} ,X) = \left \{f_{n}*x\mid x \in \mathscr{F}(\mathbb{R},X) \right \}, n \in \mathbb{N}$, witn the norm $\left \| f_{n}*x \right \|_{\mathscr{F}^{n}} = \left \| f_{n}*x \right \|_{\mathscr{F}} + \left \| x \right \|_{\mathscr{F}}$, where $ f_{n} \in L^{1}(\mathbb{R})$, $f_{n} = t^{n}e^{-t}/n!$ for $t > 0$ and $f(t) = 0$ for $ t \leq 0$. By the symbol $ \mathscr{F}^{-n}( \mathbb{R} ,X)$ for any $n \in \mathbb{N}$ we denote a harmonic distribution space defined as linear subspace of ${S}'(\mathbb{R},X)$ of distributions $\Phi \in {S}'(\mathbb{R},X)$ such that there is a function $\varphi$ from corresponding homogeneous space $ \mathscr{F}( \mathbb{R} ,X)$ defined by $\Phi = (D + I)^{n}\varphi $ and $\left \| \Phi \right \| = \left \| \varphi \right \|$. It should be noted that $\varphi = f_{n} * \Phi$. Further we consider all function and distribution spaces $ \mathscr{F}^{n}( \mathbb{R} ,X)$ with $n \in \mathbb{Z}$ as harmonic distribution spaces and prove that every such space is isometrically isomorphic to the corresponding homogeneous space $ \mathscr{F}( \mathbb{R} ,X)$. Every harmonic space $ \mathscr{F}^{n}( \mathbb{R} ,X)$ with $n \in \mathbb{Z}$ is proved to be a Banach $L^{1}(\mathbb{R})$-module with shift operators group and a structure endowed by a convolution with a function from $L^{1}(\mathbb{R})$.

On the basis of definitions of slowly varying and almost periodic at infinity functions from $ \mathscr{F}( \mathbb{R} ,X)$ we design the concept of slowly varying and almost periodic at infinity distributions from harmonic spaces. By means of the methods of abstract harmonic analysis we study the properties of those distributions. For almost periodic at infinity distributions we construct Fourier series and study their properties. The results of the article are obtained with essential use of methods of isometric representations and Banach modules theories.

Keywords: slowly varying at infinity function, almost periodic at infinity function, homogeneous
space, Banach space, distribution of slow growth, Fourier series