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The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences
Authors:
A. Bellow and V. Losert
Journal:
Trans. Amer. Math. Soc. 288 (1985), 307-345
MSC:
Primary 28D05
MathSciNet review:
773063
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Abstract: In this paper on the weighted pointwise ergodic theorem we bring together a substantial amount of previous work by a number of mathematicians and integrate it into a systematic consistent scheme; we also bring some original contributions to the subject which extend its boundaries and suggest further avenues of research. The paper is divided into six sections. The most significant new results are contained in and .
- [1]
Hirotada
Anzai and Shizuo
Kakutani, Bohr compactifications of a locally compact Abelian
group. I, Proc. Imp. Acad. Tokyo 19 (1943),
476–480. MR 0015122
(7,374e)
- [2]
J.
R. Baxter and J.
H. Olsen, Weighted and subsequential ergodic theorems, Canad.
J. Math. 35 (1983), no. 1, 145–166. MR 685822
(84g:47005), http://dx.doi.org/10.4153/CJM-1983-010-7
- [3]
Alexandra
Bellow, Sur la structure des suites “mauvaises
universelles” en théorie ergodique, C. R. Acad. Sci.
Paris Sér. I Math. 294 (1982), no. 1,
55–58 (French, with English summary). MR 651075
(84f:28012)
- [4]
Alexandra
Bellow, On “bad universal” sequences in ergodic theory.
II, Measure theory and its applications (Sherbrooke, Que., 1982)
Lecture Notes in Math., vol. 1033, Springer, Berlin, 1983,
pp. 74–78. MR 729527
(85m:28015), http://dx.doi.org/10.1007/BFb0099847
- [5]
A.
Bellow and V.
Losert, On sequences of density zero in ergodic theory,
Conference in modern analysis and probability (New Haven, Conn., 1982),
Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984,
pp. 49–60. MR 737387
(86c:28034), http://dx.doi.org/10.1090/conm/026/737387
- [6]
Jean-Paul
Bertrandias, Suites pseudo-aléatoires et critères
d’équirépartition modulo un, Compositio Math.
16 (1964), 23–28 (1964) (French). MR 0170880
(30 #1115)
- [7]
-, Espaces des fonctions bornées et continues en moyenne asymptotique d'ordre
, Bull. Soc. Math. France 5 (1966), 1-106.
- [8]
A.
S. Besicovitch, Almost periodic functions, Dover Publications
Inc., New York, 1955. MR 0068029
(16,817a)
- [9]
J.
R. Blum and R.
Cogburn, On ergodic sequences of
measures, Proc. Amer. Math. Soc. 51 (1975), 359–365. MR 0372529
(51 #8736), http://dx.doi.org/10.1090/S0002-9939-1975-0372529-1
- [10]
J.
R. Blum and J.
I. Reich, The individual ergodic theorem for
𝑝-sequences, Israel J. Math. 27 (1977),
no. 2, 180–184. MR 0442192
(56 #578)
- [11]
J.
R. Blum and J.
I. Reich, Strongly ergodic sequences of integers
and the individual ergodic theorem, Proc. Amer.
Math. Soc. 86 (1982), no. 4, 591–595. MR 674086
(84d:28020), http://dx.doi.org/10.1090/S0002-9939-1982-0674086-2
- [12]
Harald
Bohr, Almost Periodic Functions, Chelsea Publishing Company,
New York, N.Y., 1947. MR 0020163
(8,512a)
- [13]
A.
Brunel and M.
Keane, Ergodic theorems for operator sequences, Z.
Wahrscheinlichkeitstheorie und Verw. Gebiete 12 (1969),
231–240. MR 0268934
(42 #3831)
- [14]
Jean-Pierre
Conze, Convergence des moyennes ergodiques pour des
sous-suites, Contributions au calcul des probabilités, Soc.
Math. France, Paris, 1973, pp. 7–15. Bull. Soc. Math. France,
Mém. No. 35 (French). MR 0453975
(56 #12226)
- [15]
Jean
Coquet, Teturo
Kamae, and Michel
Mendès France, Sur la mesure spectrale de certaines suites
arithmétiques, Bull. Soc. Math. France 105
(1977), no. 4, 369–384 (French, with English summary). MR 0472749
(57 #12439)
- [16]
C.
Corduneanu, Almost periodic functions, Interscience Publishers
[John Wiley & Sons], New York-London-Sydney, 1968. With the
collaboration of N. Gheorghiu and V. Barbu; Translated from the Romanian by
Gitta Bernstein and Eugene Tomer; Interscience Tracts in Pure and Applied
Mathematics, No. 22. MR 0481915
(58 #2006)
- [17]
N. Dunford and J. T. Schwartz, Linear operators. I, Wiley, New York, 1958.
- [18]
W.
F. Eberlein, Abstract ergodic theorems and weak
almost periodic functions, Trans. Amer. Math.
Soc. 67 (1949),
217–240. MR 0036455
(12,112a), http://dx.doi.org/10.1090/S0002-9947-1949-0036455-9
- [19]
W.
F. Eberlein, The point spectrum of weakly almost periodic
functions, Michigan Math. J. 3 (1955–56),
137–139. MR 0082627
(18,583b)
- [20 E]
Erling
Følner, On the dual spaces of the Besicovitch almost
periodic spaces, Danske Vid. Selsk. Mat.-Fys. Medd.
29 (1954), no. 1, 27. MR 0065813
(16,490b)
- [21]
Maurice
Fréchet, Les fonctions asymptotiquement presque-periodiques
continues, C. R. Acad. Sci. Paris 213 (1941),
520–522 (French). MR 0009062
(5,96b)
- [22]
Maurice
Fréchet, Les fonctions asymptotiquement
presque-périodiques, Revue Sci. (Rev. Rose Illus.)
79 (1941), 341–354 (French). MR 0013242
(7,127e)
- [23]
Harry
Furstenberg, Poincaré recurrence and number
theory, Bull. Amer. Math. Soc. (N.S.)
5 (1981), no. 3,
211–234. MR
628658 (83d:10067), http://dx.doi.org/10.1090/S0273-0979-1981-14932-6
- [24]
K.
de Leeuw and I.
Glicksberg, Almost periodic functions on semigroups, Acta
Math. 105 (1961), 99–140. MR 0131785
(24 #A1633)
- [25]
Colin
C. Graham and O.
Carruth McGehee, Essays in commutative harmonic analysis,
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Science], vol. 238, Springer-Verlag, New York, 1979. MR 550606
(81d:43001)
- [26]
Miguel
de Guzmán, Real variable methods in Fourier analysis,
North-Holland Mathematics Studies, vol. 46, North-Holland Publishing
Co., Amsterdam, 1981. Notas de Matemática [Mathematical Notes], 75.
MR 596037
(83j:42019)
- [27]
Paul
R. Halmos and John
von Neumann, Operator methods in classical mechanics. II, Ann.
of Math. (2) 43 (1942), 332–350. MR 0006617
(4,14e)
- [28]
Alexandra
Ionescu Tulcea and Cassius
Ionescu Tulcea, Abstract ergodic theorems, Trans. Amer. Math. Soc. 107 (1963), 107–124. MR 0150611
(27 #606), http://dx.doi.org/10.1090/S0002-9947-1963-0150611-8
- [29]
Ulrich
Krengel, On the individual ergodic theorem for subsequences,
Ann. Math. Statist. 42 (1971), 1091–1095. MR 0283172
(44 #405)
- [30]
L.
Kuipers and H.
Niederreiter, Uniform distribution of sequences,
Wiley-Interscience [John Wiley & Sons], New York, 1974. Pure and
Applied Mathematics. MR 0419394
(54 #7415)
- [31]
Viktor
Losert, A class of sequences with a strong average property,
Adv. in Math. 55 (1985), no. 3, 217–223. MR 778961
(86e:28022), http://dx.doi.org/10.1016/0001-8708(85)90090-8
- [32]
D.
Newton and W.
Parry, On a factor automorphism of a normal dynamical system,
Ann. Math. Statist 37 (1966), 1528–1533. MR 0206209
(34 #6028)
- [33]
John
C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116–136. MR 0047262
(13,850e), http://dx.doi.org/10.1090/S0002-9904-1952-09580-X
- [34]
Jakob
I. Reich, On the individual ergodic theorem for subsequences,
Ann. Probability 5 (1977), no. 6, 1039–1046. MR 0444906
(56 #3252)
- [35]
V.
A. Rohlin, On the fundamental ideas of measure theory, Amer.
Math. Soc. Translation 1952 (1952), no. 71, 55. MR 0047744
(13,924e)
- [36]
C.
Ryll-Nardzewski, Topics in ergodic theory,
Probability—Winter School (Proc. Fourth Winter School, Karpacz,
1975), Springer, Berlin, 1975, pp. 131–156. Lecture Notes in
Math., Vol. 472. MR 0390177
(52 #11003)
- [37]
W. A. Veech, Commentary on
, N. Wiener: Collected Works, Vol. 1, Mathematicians of Our Time, M.I.T. Press, Cambridge, Mass., 1976.
- [38]
Peter
Walters, An introduction to ergodic theory, Graduate Texts in
Mathematics, vol. 79, Springer-Verlag, New York, 1982. MR 648108
(84e:28017)
- [39]
Norbert
Wiener, The Fourier integral and certain of its applications,
Cambridge Mathematical Library, Cambridge University Press, Cambridge,
1988. Reprint of the 1933 edition; With a foreword by Jean-Pierre Kahane.
MR 983891
(90b:42022)
- [40]
Norbert
Wiener and Aurel
Wintner, Harmonic analysis and ergodic theory, Amer. J. Math.
63 (1941), 415–426. MR 0004098
(2,319b)
- [41]
Norbert
Wiener and Aurel
Wintner, On the ergodic dynamics of almost periodic systems,
Amer. J. Math. 63 (1941), 794–824. MR 0006618
(4,15a)
- [42]
R. L. Adler, Ed., Ergodic theory and applications, Amer. Math. Soc. Summer Research Conf. (June 13-19, 1982, Univ. of New Hampshire, Durham), Amer. Math. Soc., Providence, R.I. (Informal Proceedings).
- [43]
Ulrich
Krengel, Ergodic theorems, de Gruyter Studies in Mathematics,
vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement
by Antoine Brunel. MR 797411
(87i:28001)
- [1]
- H. Anzai and S. Kakutani, Bohr compactifications of a locally compact abelian group. I, II, Proc. Imp. Acad. (Tokyo) 19 (1943), 476-480, 533-539. MR 0015122 (7:374e)
- [2]
- J. R. Baxter and J. H. Olsen, Weighted and subsequential ergodic theorems, Canad. J. Math. 35 (1983), 145-166. MR 685822 (84g:47005)
- [3]
- A. Bellow, Sur la structure des suites "mauvaises universelles" en théorie ergodique, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 55-58. MR 651075 (84f:28012)
- [4]
- -, On "bad universal" sequences in ergodic theory (II), Proc. Univ. Sherbrooke Workshop on Measure Theory and Appl. (June 1982), Lecture Notes in Math., vol. 1033, Springer-Verlag, Berlin and New York, 1983. MR 729527 (85m:28015)
- [5]
- A. Bellow and V. Losert, On sequences of density zero in ergodic theory, Proc. Conf. on Modern Analysis and Probability Honoring Prof. Shizuo Kakutani, Contemporary Math. Ser., vol. 26, Amer. Math. Soc., Providence, R.I., 1984, pp. 49-60. MR 737387 (86c:28034)
- [6]
- J. P. Bertrandias, Suites pseudo-aléatoires et critères d'équirépartition module
, Compositio Math. (Groningen) 16 (1964), 23-28. MR 0170880 (30:1115)
- [7]
- -, Espaces des fonctions bornées et continues en moyenne asymptotique d'ordre
, Bull. Soc. Math. France 5 (1966), 1-106.
- [8]
- A. S. Besicovitch, Almost periodic functions, 2nd ed., Dover, New York, 1954. MR 0068029 (16:817a)
- [9]
- J. R. Blum and R. Cogburn, on ergodic sequences of measures, Proc. Amer. Math. Soc. 51 (1975), 359-365. MR 0372529 (51:8736)
- [10]
- J. R. Blum and J. I. Reich, The Individual Ergodic Theorem for
-sequences, Israel J. Math. 27 (1977), 180-184. MR 0442192 (56:578)
- [11]
- -, Strongly ergodic sequences of integers and the Individual Ergodic Theorem, Proc. Amer. Math. Soc. 86 (1982), 591-595. MR 674086 (84d:28020)
- [12]
- H. Bohr, Almost periodic functions, Chelsea, New York, 1947. MR 0020163 (8:512a)
- [13]
- A. Brunel and M. Keane, Ergodic theorems for operator sequences, Z. Wahrsch. Verw. Gebiete 12 (1969), 231-240. MR 0268934 (42:3831)
- [14]
- J. P. Conze, Convergence des moyennes ergodiques pour des sous-suites, Bull. Soc. Math. France 35 (1973), 7-15. MR 0453975 (56:12226)
- [15]
- J. Coquet, T. Kamae and M. Mendès France, Sur la mesure spectrale de certaines suites arithmétiques, Bull. Soc. Math. France 105 (1977), 369-384. MR 0472749 (57:12439)
- [16]
- C. Corduneanu, Almost periodic functions, Wiley-Interscience, New York, 1968. MR 0481915 (58:2006)
- [17]
- N. Dunford and J. T. Schwartz, Linear operators. I, Wiley, New York, 1958.
- [18]
- W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217-240. MR 0036455 (12:112a)
- [19]
- -, The point spectrum of weakly almost periodic functions, Michigan J. Math. 3 (1956), 137-139. MR 0082627 (18:583b)
- [20 E]
- Følner, On the dual spaces of the Besicovitch almost periodic spaces, Mat.-Fys. Medd. Danske Vid. Selsk. 29 (1954). MR 0065813 (16:490b)
- [21]
- M. Fréchet, Les fonctions asymptotiquement presque périodiques continues, C. R. Acad. Sci. Paris Sér. I Math. 213 (1941), 520-522. MR 0009062 (5:96b)
- [22]
- -, Les fonctions asymptotiquement presque-périodiques, Rev. Sci. 79 (1941), 341-354. MR 0013242 (7:127e)
- [23]
- H. Furstenberg, Poincaré recurrence and number theory, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 211-234. MR 628658 (83d:10067)
- [24]
- I. Glicksberg and K. de Leeuw, Almost periodic functions on semigroups, Acta Math. 105 (1961), 99-140. MR 0131785 (24:A1633)
- [25]
- C. C. Graham and O. C. McGehee, Essays in commutative harmonic analysis, Grundlehren Math. Wiss., Springer-Verlag, Berlin and New York, Vol. 238, 1979. MR 550606 (81d:43001)
- [26]
- M. de Guzmán, Real variable methods in Fourier analysis, North-Holland Math. Stud., Vol. 46, Amsterdam, New York, 1981. MR 596037 (83j:42019)
- [27]
- P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. II, Ann. of Math. (2) 43 (1942), 332-350. MR 0006617 (4:14e)
- [28]
- A. Ionescu Tulcea and C. Ionescu Tulcea, Abstract ergodic theorems, Trans. Amer. Math. Soc. 107 (1963), 107-124. MR 0150611 (27:606)
- [29]
- U. Krengel, On the Individual Ergodic Theorem for subsequences, Ann. Math. Statist. 42 (1971), 1091-1095. MR 0283172 (44:405)
- [30]
- L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience, New York, 1974. MR 0419394 (54:7415)
- [31]
- V. Losert, On a class of sequences with a strong average property, Adv. in Math. (to appear). MR 778961 (86e:28022)
- [32]
- D. Newton and W. Parry, On a factor automorphism of a normal dynamical system, Ann. Math. Statist. 37 (1966), 1528-1533. MR 0206209 (34:6028)
- [33]
- J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116-136. MR 0047262 (13:850e)
- [34]
- J. I. Reich, On the Individual Ergodic Theorem for subsequences, Ann. Probab. 5 (1977), 1039-1046. MR 0444906 (56:3252)
- [35]
- V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Transl., Vol. 10, No. 1 (1962), 1-54. MR 0047744 (13:924e)
- [36]
- C. Ryll-Nardzewski, Topics in ergodic theory, Lecture Notes in Math., Vol. 472, Springer-Verlag, 1975, pp. 131-156. MR 0390177 (52:11003)
- [37]
- W. A. Veech, Commentary on
, N. Wiener: Collected Works, Vol. 1, Mathematicians of Our Time, M.I.T. Press, Cambridge, Mass., 1976.
- [38]
- P. Walters, An introduction to ergodic theory, Graduate Texts in Math. No. 79, Springer-Verlag, New York, 1982. MR 648108 (84e:28017)
- [39]
- N. Wiener, Fourier integral and certain of its applications, Cambridge Univ. Press, London and New York, 1933. MR 983891 (90b:42022)
- [40]
- N. Wiener and A. Wintner, Harmonic analysis and ergodic theory, Amer. J. Math. 63 (1941), 415-426. MR 0004098 (2:319b)
- [41]
- -, On the ergodic dynamics of almost periodic systems, Amer. J. Math. 63 (1941), 794-824. MR 0006618 (4:15a)
- [42]
- R. L. Adler, Ed., Ergodic theory and applications, Amer. Math. Soc. Summer Research Conf. (June 13-19, 1982, Univ. of New Hampshire, Durham), Amer. Math. Soc., Providence, R.I. (Informal Proceedings).
- [43]
- U. Krengel, Ergodic theorems, monograph, de Gruyter Stud. in Math., de Gruyter, Berlin (to appear). MR 797411 (87i:28001)
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DOI:
http://dx.doi.org/10.1090/S0002-9947-1985-0773063-8
PII:
S 0002-9947(1985)0773063-8
Keywords:
Automorphism,
ergodic,
discrete spectrum,
countable Lebesgue spectrum,
-automorphism,
"good universal weight",
positive definite function,
affinity of two probability measures (=Hellinger integral),
correlation of a sequence,
spectral measure corresponding to a sequence,
Besicovitch class ,
almost periodic functions (in the sense of Bohr, Weyl, Eberlein),
Bochner-Fejér polynomial,
strictly -stable dynamical system,
"uniform sequence",
Bohr compactification,
sequence that satisfies a "uniform order conditin on ",
"saturating sequence",
Weak Maximal Inequality,
"bad universal sequence",
"block sequence",
lacunary sequence,
"good universal sequence" of density zero
Article copyright:
© Copyright 1985 American Mathematical Society
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