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Some problems concerning different types of vector valued almost periodic functions. (English) Zbl 0828.43004

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)

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.
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