zbMATH — the first resource for mathematics

Vector-valued pseudo almost periodic functions. (English) Zbl 0901.42005
The author deals with a new generalization of (vector-valued) almost periodic functions to (vector-valued) pseudo almost periodic functions. Many well-known functions connected with the notion of almost periodic functions are contained as special cases in the class of these pseudo almost periodic functions.
Definition 5 defines a pseudo periodic function using a generalized almost periodicity and a generalized uniform continuity.
The pseudo almost periodic functions remind the former generalization of almost periodic functions to asymptotically almost periodic functions with their unique decomposition as a sum of an almost periodic function and a limit zero perturbation. The main result of this paper consists in Theorem 11 which shows that any pseudo almost periodic function has a unique decomposition as a sum of an almost periodic function and an ergodic perturbation.
The interesting Example 14 illustrates the fact that the generalized almost periodicity of a pseudo almost periodic function does not guarantee its generalized uniform continuity.
The space of pseudo almost periodic functions is a Banach space.
Reviewer: A.Fischer (Praha)

42A75 Classical almost periodic functions, mean periodic functions
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
Full Text: DOI EuDML
[1] L. Amerio and G. Prouse: Almost-Periodic Functions and Functional Equations. Van Nostrand, New York, 1971. · Zbl 0215.15701
[2] J.F. Berglund, H.D. Junghenn and P. Milnes: Analysis on Semigroups: Function Spaces, Compactifications, Representations. Wiley, New York, 1989. · Zbl 0727.22001
[3] A. S. Besicovitch: Almost Periodic Functions. Dover, New York, 1954. · Zbl 0065.07102
[4] H.A. Bohr: Almost Periodic Functions. Chelsea, New York, 1951. · Zbl 0005.20303
[5] C. Corduneanu: Almost Periodic Functions. Chelsea, New York, 2nd, 1989. · Zbl 0672.42008
[6] K. de Leeuw and I. Glicksberg: Applications of almost periodic compactifications. Acta Math. 105 (1961), 63-97. · Zbl 0104.05501
[7] S. Goldberg and P. Irwin: Weakly almost periodic vector valued functions. Dissertationes Math. 157 (1979). · Zbl 0414.43007
[8] B. M. Levitan and V. V. Zhikov: Almost periodic functions and differential equations. Cambridge University Press, New York, 1982. · Zbl 0499.43005
[9] P. Milnes: On vector-valued weakly almost periodic functions. J. London Math. Soc. (2) 22 (1980), 467-472. · Zbl 0456.43003
[10] W.M. Ruess and W.H. Summers: Compactness in spaces of vector valued continuous functions and asymptotic almost periodicity. Math. Nachr. 135 (1988), 7-33. · Zbl 0666.46007
[11] W.M. Ruess and W.H. Summers: Integration of asymptotically almost periodic functions and weak asymptotic almost periodicity. Dissertationes Math. 279 (1989). · Zbl 0668.43005
[12] S. Zaidman: Almost-Periodic Functions in Abstract Spaces. Pitman, London, 1985. · Zbl 0648.42006
[13] C. Zhang: Pseudo almost periodic solutions of some differential equations. J. Math. Anal. Appl. 181 (1994), 62-76. · Zbl 0796.34029
[14] C. Zhang: Pseudo almost periodic solutions of some differential equations, II. J. Math. Anal. Appl. 192 (1995), 543-561. · Zbl 0826.34040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.