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Abstract ergodic theorems and weak almost periodic functions
Author:
W. F. Eberlein
Journal:
Trans. Amer. Math. Soc. 67 (1949), 217-240
MSC:
Primary 46.3X
MathSciNet review:
0036455
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Additional Information
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Alaoglu and G.
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S. Phillips, On weakly compact subsets of a Banach space,
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A. Raikov, Harmonic analysis on commutative groups with the Haar
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14 (1945), 86 (Russian, with English summary). MR 0017285
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Šmulian, On some problems of the functional analysis,
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Béla
de Sz. Nagy, On uniformly bounded linear transformations in Hilbert
space, Acta Univ. Szeged. Sect. Sci. Math. 11 (1947),
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(9,191b)
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W. Tukey, Convergence and Uniformity in Topology, Annals of
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Kôsaku
Yosida and Shizuo
Kakutani, Operator-theoretical treatment of Markoff’s process
and mean ergodic theorem, Ann. of Math. (2) 42
(1941), 188–228. MR 0003512
(2,230e)
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A. Zygmund, Trigonometrical series, Warsaw, 1935.
- [1]
- L. Alaoglu and G. Birkhoff, General ergodic theorems. Ann. of Math. (2) vol. 41 (1940) pp. 293-309. MR 0002026 (1:339a)
- [2]
- S. Banach, Théorie des opérations linéaires, Warsaw, 1932.
- [3]
- A. S. Besicovitch, Almost periodic functions, Cambridge, England, 1932.
- [4]
- S. Bochner, Vorlesungen über Fouriersche Integrale, Leipzig, 1932.
- [5]
- H. Bohr and E. Følner, On some types of functional spaces, Acta Math. vol. 76 (1945) pp. 31-155. MR 0013443 (7:154f)
- [6]
- D. G. Bourgin, Linear topological spaces, Amer. J. Math. vol. 65 (1943) pp. 637-659. MR 0009104 (5:103a)
- [7]
- J. W. Calkin, Abstract symmetric boundary conditions, Trans. Amer. Math. Soc. vol. 45 (1939) pp. 369-442. MR 1501997
- [8]
- M. M. Day, Operations in Banach spaces, Trans. Amer. Math. Soc. vol. 51 (1942) pp. 583-608. MR 0006613 (4:14a)
- [9]
- -, Ergodic theorems for Abelian semi-groups, Trans. Amer. Math. Soc. vol. 51 (1942) pp. 399-412. MR 0006614 (4:14b)
- [10]
- (a) N. Dunford, Spectral theory. I. Convergence to projections, Trans. Amer. Math. Soc. vol. 54 (1943) pp. 185-217. MR 0008642 (5:39c)
- 1.
- (b) -, Spectral theory, Bull. Amer. Math. Soc. vol. 49 (1943) pp. 637-651. MR 0008643 (5:39d)
- [11]
- N. Dunford and D. S. Miller, On the ergodic theorem, Trans. Amer. Math. Soc. vol. 60 (1946) pp. 538-549. MR 0018359 (8:280b)
- [12]
- N. Dunford and B. J. Pettis, Linear operations on summable functions, Trans. Amer. Math. Soc. vol. 47 (1940) pp. 323-392. MR 0002020 (1:338b)
- [13]
- W. F. Eberlein, Weak compactness in Banach spaces I, Proc. Nat. Acad. Sci. U.S.A. vol. 33 (1947) pp. 51-53. MR 0021239 (9:42a)
- [14]
- -, Abstract ergodic theorems, Proc. Nat. Acad. Sci. U.S.A. vol. 34 (1948) pp. 43-47. MR 0023460 (9:359c)
- [15]
- M. Eidelheit, On isomorphisms of rings of linear operators, Studia Math. vol. 9 (1940) pp. 97-105. MR 0004725 (3:51e)
- [16]
- I. Gelfand and M. Neumark, On the imbedding of normed rings into the ring of operators in Hilbert space, Rec. Math. (Math. Sbornik) N.S. vol. 12 (1943) pp. 197-213. MR 0009426 (5:147d)
- [17]
- R. Godement, Les fonctions de type positif et la théorie des groupes, Trans. Amer. Math. Soc. vol. 63 (1948) pp. 1-84. MR 0023243 (9:327b)
- [18]
- W. H. Gottschalk, Almost periodicity, equi-continuity, and total boundedness, Bull. Amer. Math. Soc. vol. 52 (1946) pp. 633-636. MR 0016551 (8:34b)
- [19]
- E. Hille, Remarks on ergodic theorems, Trans. Amer. Math. Soc. vol. 57 (1945) pp. 246-269. MR 0012212 (6:276d)
- [20]
- D. H. Hyers, Linear topological spaces, Bull. Amer. Math. Soc. vol. 51 (1945) pp. 1-21. MR 0012205 (6:275a)
- [21]
- S. Kakutani, Two fixed-point theorems concerning bicompact convex sets, Proc. Imp. Acad. Tokyo vol. 14 (1939) pp. 242-245. MR 1568507
- [22]
- -, Concrete representation of abstract
-spaces, Ann. of Math. (2) vol. 42 (1941) pp. 994-1024. MR 0005778 (3:205g)
- [23]
- Y. Kawada, Über den Mittelwert der messbaren fastperiodischen Funktionen auf einer Gruppe, Proc. Imp. Acad. Tokyo vol. 19 (1943) pp. 264-266. MR 0014098 (7:240h)
- [24]
- W. Maak, Eine neue Definition der fastperiodischen Funktionen, Abh. Math. Sem. Hamburgischen Univ. vol. 11 (1936) pp. 240-244.
- [25]
- J. von Neumann, Almost periodic functions in a group. I, Trans. Amer. Math. Soc. vol. 36 (1934) pp. 445-492. MR 1501752
- [26]
- R. S. Phillips, On weakly compact subsets of a Banach space, Amer. J. Math. 65 (1943) pp. 108-136. MR 0007938 (4:218f)
- [27]
- D. A. Raikov, Harmonic analysis on commutative groups with the Haar measure and the theory of characters (Russian), Travaux de l'Institut Mathématique Stekloff vol. 14 (1945). MR 0017285 (8:133a)
- [28]
- F. Riesz, Some mean ergodic theorems, J. London Math. Soc. vol. 13 (1938) pp. 274-278.
- [29]
- V. Smulian, On some problems of the functional analysis, C. R. (Doklady) Acad. Sci. URSS N.S. vol. 38 (1943) pp. 157-159. MR 0008637 (5:38c)
- [30]
- B. de Sz. Nagy, On uniformly bounded linear transformations in Hilbert space, Acta Univ. Szeged. vol. 11 (1947) pp. 152-157. MR 0022309 (9:191b)
- [31]
- J. W. Tukey, Convergence and uniformity in topology, Princeton, 1940. MR 0002515 (2:67a)
- [32]
- A. Weil, L'integration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles, no. 869, Paris, 1940.
- [33]
- K. Yosida and S. Kakutani, Operator-theoretical treatment of Markoff's process and mean ergodic theorem, Ann. of Math. (2) vol. 42 (1941) pp. 188-228. MR 0003512 (2:230e)
- [34]
- A. Zygmund, Trigonometrical series, Warsaw, 1935.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9947-1949-0036455-9
PII:
S 0002-9947(1949)0036455-9
Article copyright:
© Copyright 1949 American Mathematical Society
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