Generalized almost periodicity in groups

Davis, Henry W.

doi:10.1090/S0002-9947-1974-0348400-1

Generalized almost periodicity in groups
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Trans. Amer. Math. Soc. 191 (1974), 329-352 Request permission

Abstract:

A module of almost periodic functions on a group is closed with respect to a quite general seminorm. The new space of functions is characterized in terms of the internal properties of its members. This yields new characterizations of Besicovitch and Weyl almost periodic functions in a variety of group-theoretic settings. Eberlein’s theorem that weakly almost periodic functions on the real line are Weyl almost periodic is extended to locally compact groups.

References

Hirotada Anzai and Shizuo Kakutani, Bohr compactifications of a locally compact Abelian group. I, Proc. Imp. Acad. Tokyo 19 (1943), 476–480. MR 15122
John F. Berglund, On extending almost periodic functions, Pacific J. Math. 33 (1970), 281–289. MR 412742, DOI 10.2140/pjm.1970.33.281
J. F. Berglund and K. H. Hofmann, Compact semitopological semigroups and weakly almost periodic functions, Lecture Notes in Mathematics, No. 42, Springer-Verlag, Berlin-New York, 1967. MR 0223483, DOI 10.1007/BFb0073920
A. S. Besicovitch, Almost periodic functions, Dover Publications, Inc., New York, 1955. MR 0068029
Abdallah N. Dabboucy and Henry W. Davis, A new characterization of Besicovitch almost periodic functions, Math. Scand. 28 (1971), 341–354 (1972). MR 326288, DOI 10.7146/math.scand.a-11029
Henry W. Davis, On the mean value of Haar measurable almost periodic functions, Duke Math. J. 34 (1967), 201–214. MR 212120
Henry W. Davis, Stepanoff and Weyl $\textrm {AP}$ functions on locally compact groups, Duke Math. J. 34 (1967), 535–548. MR 215944
Henry W. Davis, On completing the von Neumann almost periodic functions, Duke Math. J. 35 (1968), 199–216. MR 222564
Henry W. Davis, On a formula for Haar measure in compact groups, Math. Scand. 30 (1972), 51–61. MR 324310, DOI 10.7146/math.scand.a-11063
W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217–240. MR 36455, DOI 10.1090/S0002-9947-1949-0036455-9
W. R. Emerson and F. P. Greenleaf, Covering properties and Følner conditions for locally compact groups, Math. Z. 102 (1967), 370–384. MR 220860, DOI 10.1007/BF01111075
Erling Følner, On the structure of generalized almost periodic functions, Danske Vid. Selsk. Mat.-Fys. Medd. 21 (1945), no. 11, 30. MR 16167
Erling Følner, $W$-almost-periodic functions on arbitrary groups, Mat. Tidsskr. B 1946 (1946), 153–162 (Danish). MR 16422
Erling Følner, Besicovitch almost periodic functions in arbitrary groups, Math. Scand. 5 (1957), 47–53. MR 91379, DOI 10.7146/math.scand.a-10488
Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
A. Grothendieck, Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), 168–186 (French). MR 47313, DOI 10.2307/2372076
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496, DOI 10.1007/978-1-4419-8638-2
E. R. van Kampen, Almost periodic functions and compact groups, Ann. of Math. (2) 37 (1936), no. 1, 78–91. MR 1503270, DOI 10.2307/1968688
Wilhelm Maak, Fastperiodische Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1950 (German). MR 0041961, DOI 10.1007/978-3-642-52797-5