Bohr conditions and almost periodic means in quasi-complete spaces

Document Type : Dedicated to Prof. A. T.-M. Lau

Authors

Department of Mathematics, ‎University of Windsor, Ontario, Canada.

Abstract

We study Bohr conditions for functions on topological groups taking values in locally convex spaces. We show that functions satisfying Bohr conditions are uniformly continuous. We show that in quasi-complete spaces, Bohr conditions are equivalent to Bochner's characterizations of almost periodicity. We prove the existence of invariant mean for almost periodic functions with values in quasi-complete spaces.

Keywords

Main Subjects


[1] S. Bochner, Abstrakte Fastperiodische Funktionen. (German), Acta Math. 61 (1933), no. 1, 149–184.
[2] S. Bochner and J. von Neumann, Almost periodic functions in groups, II, Trans. Amer. Math. Soc. 37 (1935), no. 1, 21–50.
[3] H. Bohr, Zur theorie der fast periodischen funktionen. (German) I, Eine Verallgemeinerung der Theorie der Fourierreihen, Acta. Math. 45 (1925), no. 1, 29–127.
[4] N. Bourbaki, Elements of Mathematics: Topological Vector Spaces, Chapters 1–5, Springer, New York, 2003.
[5] C. Chou and A. T.-M. Lau, Vector-valued invariant means on spaces of bounded operators associated to a locally compact group, Illinois J. Math. 45 (2001), no. 2, 581–602.
[6] P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935), no. 1, 26–30.
[7] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1, Second Edition, Springer-Verlag, Berlin, 1979.
[8] W. Maak, Fastperiodische Funktionen, Springer, Berlin, 1950.
[9] G. M. N’Guérékata, Almost periodicity in linear topological spaces and applications to abstract differential equations, Internat. J. Math. Math. Sci. 7 (1984), no. 3, 529–540.
[10] G. M. N’Guérékata, Almost Periodic and Almost Automorphic Functions in Abstract Spaces, Second Edition, Springer, New York, 2021.
[11] B. Simon, Operator Theory, A Comprehensive Course in Analysis, Part 4, Amer. Math. Soc., Providence, Rhode Island, 2015.
[12] F. Trèves, Topological Vector Spaces, Distributions and Kernels, Dover Publications, New York, 1995.
[13] J. von Neumann, On complete topological spaces, Trans. Amer. Math. Soc. 37 (1935), no. 1, 1–20.