Basit, Bolis Some problems concerning different types of vector valued almost periodic functions. (English) Zbl 0828.43004 Diss. Math. 338, 26 p. (1995). Collecting and unifying earlier results in this field, theorems of the following type are treated: If \(\varphi : \mathbb{R} \to\) Banach space \(X\) is almost periodic (a.p.) in some sense, with \(\psi (t) : = \int^t_0 \varphi ds\) ergodic or \(0 \notin\) Beurling spectrum of \(\varphi\), then \(\psi\) is a.p.: With \(J = \mathbb{R}^+\) or \(\mathbb{R}\), \(C_u\) resp. \(C_{ub} (J,X)\) containing all uniformly continuous resp. uniformly continuous and bounded \(\varphi : J \to X\), an \(A \subset C_{ub} : = C_{ub} (J,X)\) is called a \(\Lambda\)-class iff \(A\) is a closed linear subspace of \(C_{ub}\), \(e^{i \omega t} \varphi \in A\) for \(\varphi \in A\), \(\omega \in \mathbb{R}\), constants \(\in A\), and \((*)\) \(\psi (a + \cdot) |J \in A\) for all \(a \in \mathbb{R}\) and \(\psi \in C_{ub} (\mathbb{R},X)\) with \(\psi |J \in A\); \(A\) is a \(\Pi\)-class, if it satisfies these conditions, but with \((*)\) replaced by \((\Pi)\): to \(\varphi \in A\) there exists \(\psi \in C_{ub} (\mathbb{R},X)\) with \(\psi |J = \varphi\) and \(\psi (a + \cdot) |J \in A\), \(a \in \mathbb{R}\). A closed linear translation invariant subspace of \(C_{ub}\) containing the constants is called a TICSC-class. Bochner’s theorem is generalized thus: If \(A\) is a translation invariant closed subspace of \(C_{ub}\) resp. a \(\Lambda \)- or \(\Pi\)-class, \(\varphi \in A\) and \(\varphi' \in C_u\), then \(\varphi' \in A\). Examples: A.p., almost automorphic functions, the span of translates of a recurrent \(\varphi\), and their asymptotic versions are \(\Pi\)-classes, for \(J = \mathbb{R}\) also \(\Lambda\)-classes. Also Eberlein-a.p. \(\varphi : J \to X\) \((\{\varphi (a + \cdot) : a \in J\}\) is weakly relatively compact in \(C_b = \) bounded continuous functions) and totally ergodic functions form \(\Lambda\)-classes in \(C_{ub} \); here \(\varphi \in C_{ub}\) is ergodic iff \((1/T) \int^T_0 \varphi (t + s) ds\) converges \(t\)-uniformly to a constant (for \(J = \mathbb{R} : (1/2T) \int^T_{- T} \ldots)\); \(\varphi\) is totally ergodic if all \(e^{i \omega \cdot} \varphi\) are ergodic, \(\omega \in \mathbb{R}\). If \(\psi \in C_{ub}\), \(\psi' \in C_u\), then \(\psi'\) is ergodic; e.g. \(\psi (t) = \int^t_0 \varphi ds\) bounded, \(\varphi \in C_{ub \cdot} \text{ \{a.p.\}} \subset \text{\{as.a.p.\}} \subset \text{\{Eberlein-a.p.\}} \subset \text{ \{tot.erg.\}} \subset \text{\{ergodic\}}\). If \(\varphi \in A \subset C_{ub}\), \(A\) a TICSC-class, \(\psi (t) : = \int^t_0 \varphi ds\) is ergodic or \(\int^t_0 \psi ds \in C_{ub}\), then \(\psi \in A\); more generally, if \(g : J \to X\) is ergodic and all \(g(a + \cdot) - g(\cdot) \in A\), \(a \in J\), again \(g \in A\); this is also true if \(A = \{\varphi\) as. a.p. in the weak topology}, with corresponding weak ergodicity. – For a \(\Lambda\)-class \(A\) and \(\varphi \in C_{ub}\), spectrum \(\sigma_A (\varphi) : = \{\omega \in \mathbb{R} : \widehat f (\omega) = 0\) for all \(f \in I_A (\varphi)\}\), with \(I_A (\varphi) : = \{f \in L^1 : = L^1 (\mathbb{R}, C) : \varphi*f |J \in A\}\), a closed ideal of \(L^1\). Always \(\sigma_A (\varphi) \subset\) Beurling spectrum \(\sigma_0 (\varphi) = \sigma_{\{0\}} (\varphi)\). If \(A\) is a \(\Lambda\)-class, \(\varphi \in C_{ub} (\mathbb{R}, X)\), then \(\sigma_A (\varphi) = \emptyset \) iff \(\varphi |J \in A\). \(\sigma_A (\varphi) \subset \{0\}\) iff all \(\varphi (a + \cdot) - \varphi (\cdot) \in A\), \(a \in J\); special case: \(\varphi\) ergodic, \(\sigma_A (\varphi) \subset \{0\}\), then \(\varphi \in A\). – Generalized Favard theorem: If \(\varphi \in C_{ub} (\mathbb{R},X)\), \(A\) is a TICSC-class, \(0 \notin \sigma_0 (\varphi)\), \(\varphi |J \in A\), then \(\int^t_0 \varphi ds \in A\). \(\varphi \in C_{ub} (\mathbb{R}, X)\), \(A\) \(\Lambda\)-class, \(\sigma_A (\varphi)\) closed countable, \(\varphi |J\) totally ergodic, then \(\varphi \in A\). – This is applied to linear integro-differential difference equations: If the right hand side is in some \(\Lambda\)-class \(A\), then if an \(X\)-valued solution on \(\mathbb{R}^+\) is totally ergodic, it is in \(A\), provided the Fourier transform of the characteristic function is analytic on \(\mathbb{R}\). (Communication of the author: In equation \((*)\) on p. 22 the integral term has to be replaced by \(\int^\infty_0 g(s) \omega (s + t) ds\), with corresponding changes later, e.g. “\(\cdots + \check G* \Omega* h(t) \cdots\)” in (5.2.2); for Lemma 5.1.3 and in the proof of theorem 5.2.7 an extension of the Esclangon-Landau theorem to such differential-difference equations, due to the author, is used). Reviewer: H.Günzler (Kiel) Cited in 1 ReviewCited in 24 Documents MSC: 43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions 34K14 Almost and pseudo-almost periodic solutions to functional-differential equations 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations 34C28 Complex behavior and chaotic systems of ordinary differential equations 43A45 Spectral synthesis on groups, semigroups, etc. Keywords:almost periodic; spectrum; indefinite integral; differential-difference equation PDFBibTeX XMLCite \textit{B. Basit}, Diss. Math. 338, 26\,p. (1995; Zbl 0828.43004)