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 Uspekhi Mat. Nauk, 2005, Volume 60, Issue 3(363), Pages 97–168 (Mi umn1430)

Almost periodic functions and representations in locally convex spaces

A. I. Shtern

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Properties of diverse classes of almost periodic functions with values in locally convex spaces and of almost periodic representations on locally convex spaces are considered. The well-known criterion for the almost periodicity of weakly almost periodic group representations on Banach spaces (in terms of scalar almost periodicity) is extended to the case of weakly continuous weakly almost periodic representations on barrelled spaces in which the weakly closed convex hulls of weakly compact sets are weakly compact. Applications of this result are indicated and a survey of the current state of some other classical problems in the theory of almost periodic functions (as applied to almost periodic functions with values in locally convex spaces) and modern directions of investigation related to almost periodic functions on groups and finite-dimensional unitary representations of groups are presented. In particular, decomposition problems for weakly almost periodic representations and characterizations of diverse classes of almost periodic functions (including criteria for almost periodicity), existence problems for the mean value, countability conditions for the spectrum of a scalarly almost periodic function, theorems on the integral and the differences of almost periodic functions, and other relationships among strong, scalar, and weak almost periodicity for functions with values in locally convex spaces are treated.

DOI: https://doi.org/10.4213/rm1430

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English version:
Russian Mathematical Surveys, 2005, 60:3, 489–557

Bibliographic databases:

UDC: 517.986.63+517.986.4
MSC: Primary 43A60, 22A25; Secondary 42A75, 43A07, 22A20, 46A32, 47D03, 46A08, 22D10, 3

Citation: A. I. Shtern, “Almost periodic functions and representations in locally convex spaces”, Uspekhi Mat. Nauk, 60:3(363) (2005), 97–168; Russian Math. Surveys, 60:3 (2005), 489–557

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn1430
• https://doi.org/10.4213/rm1430
• http://mi.mathnet.ru/eng/umn/v60/i3/p97

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. I. Shtern, “Finite-dimensional quasirepresentations of connected Lie groups and Mishchenko's conjecture”, J. Math. Sci., 159:5 (2009), 653–751
2. A. I. Shtern, “Kazhdan–Milman problem for semisimple compact Lie groups”, Russian Math. Surveys, 62:1 (2007), 113–174
3. A. I. Shtern, “Duality between compactness and discreteness beyond Pontryagin duality”, Proc. Steklov Inst. Math., 271 (2010), 212–227
4. A. I. Shtern, “The structure of homomorphisms of connected locally compact groups into compact groups”, Izv. Math., 75:6 (2011), 1279–1304
5. M. I. Karakhanian, “Almost periodicity in spectral analysis representations induced by generalized shift operation”, Uch. zapiski EGU, ser. Fizika i Matematika, 2012, no. 3, 9–13
6. Khadjiev D., Cavus A., “Continuous Invariant Averagings”, Turk. J. Math., 37:5 (2013), 770–780
7. I. A. Trishina, “Pochti periodicheskie na beskonechnosti funktsii otnositelno podprostranstva integralno ubyvayuschikh na beskonechnosti funktsii”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 17:4 (2017), 402–418
8. Shtern I A., “Continuity Conditions For Finite-Dimensional Locally Bounded Representations of Connected Locally Compact Groups”, Russ. J. Math. Phys., 25:3 (2018), 345–382
9. A. G. Baskakov, V. E. Strukov, I. I. Strukova, “Harmonic analysis of functions in homogeneous spaces and harmonic distributions that are periodic or almost periodic at infinity”, Sb. Math., 210:10 (2019), 1380–1427
10. Kostic M., “Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations”, Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations, Walter de Gruyter Gmbh, 2019, 1–329
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