On some classes of almost periodic functions in abstract spaces (Q1777845)

Let \((\mathbb{E},\|\cdot \|)\) be a Banach space and let \(f:\mathbb{R}\to\mathbb{E}\) have continuous derivatives of orders \(i\leq n\) at all points \(t\in\mathbb{R}\), with finite norms \[ \| f\|_n= \sup_{t\in\mathbb{R}}\,\Biggl(\| f(t)\|+ \sum^n_{i=1}\| f^{(i)}(t)\|\Biggr). \] Then \(C^{(n)}\)-almost periodicity of \(f\) is defined by the relative density of the set \(E^{(n)}(\varepsilon, f)\) of all \((\|\cdot \|_n,\varepsilon)\)-almost periods of \(f\), like in the classical case of u.a.p.-functions. There are investigated properties of such functions, their derivatives and integrals. Special attention is paid to the superposition operator. There is also introduced and investigated the notion of asymptotically \(C^{(n)}\)-almost periodic functions.