## The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences.(English)Zbl 0619.47004

The principal object of this work is to give an extensive coherent treatment of ergodic theorems for weighted averages and ergodic theorems along subsequences. In other words, the authors study the question for which sequences $$(a_ k)$$ of complex numbers and for which operators in $$L_ p$$ the averages (1/n)$$\sum^{n}_{0}a_ kT^ kf$$ converge a.e. for all $$f\in L_ p$$. A related question is for which strictly increasing sequences $$(n_ k)$$ of integers the averages (1/n)$$\sum^{n}_{0}T^{n_ k}f$$ converge a.e. The paper also contains a number of new results, and suggests further avenues of research.
Reviewer: U.Krengel

### MSC:

 47A35 Ergodic theory of linear operators 28D05 Measure-preserving transformations
Full Text:

### References:

 [1] Hirotada Anzai and Shizuo Kakutani, Bohr compactifications of a locally compact Abelian group. I, Proc. Imp. Acad. Tokyo 19 (1943), 476 – 480. · Zbl 0063.00102 [2] J. R. Baxter and J. H. Olsen, Weighted and subsequential ergodic theorems, Canad. J. Math. 35 (1983), no. 1, 145 – 166. · Zbl 0478.47007 [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). · Zbl 0501.28009 [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. [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. · Zbl 0587.28013 [6] Jean-Paul Bertrandias, Suites pseudo-aléatoires et critères d’équirépartition modulo un, Compositio Math. 16 (1964), 23 – 28 (1964) (French). · Zbl 0207.05801 [7] -, Espaces des fonctions bornées et continues en moyenne asymptotique d’ordre $$p$$, Bull. Soc. Math. France 5 (1966), 1-106. · Zbl 0148.11701 [8] A. S. Besicovitch, Almost periodic functions, Dover Publications, Inc., New York, 1955. · Zbl 0065.07102 [9] J. R. Blum and R. Cogburn, On ergodic sequences of measures, Proc. Amer. Math. Soc. 51 (1975), 359 – 365. · Zbl 0309.43001 [10] J. R. Blum and J. I. Reich, The individual ergodic theorem for \?-sequences, Israel J. Math. 27 (1977), no. 2, 180 – 184. · Zbl 0361.28009 [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. · Zbl 0552.28014 [12] Harald Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N.Y., 1947. · Zbl 0005.20303 [13] A. Brunel and M. Keane, Ergodic theorems for operator sequences, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 12 (1969), 231 – 240. · Zbl 0187.00904 [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). · Zbl 0285.28017 [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). · Zbl 0383.10035 [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. · Zbl 0672.42008 [17] N. Dunford and J. T. Schwartz, Linear operators. I, Wiley, New York, 1958. · Zbl 0084.10402 [18] W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217 – 240. · Zbl 0034.06404 [19] W. F. Eberlein, The point spectrum of weakly almost periodic functions, Michigan Math. J. 3 (1955 – 56), 137 – 139. [20] Erling Følner, On the dual spaces of the Besicovitch almost periodic spaces, Danske Vid. Selsk. Mat.-Fys. Medd. 29 (1954), no. 1, 27. · Zbl 0059.07304 [21] Maurice Fréchet, Les fonctions asymptotiquement presque-periodiques continues, C. R. Acad. Sci. Paris 213 (1941), 520 – 522 (French). · Zbl 0026.22102 [22] Maurice Fréchet, Les fonctions asymptotiquement presque-périodiques, Revue Sci. (Rev. Rose Illus.) 79 (1941), 341 – 354 (French). · Zbl 0061.16301 [23] Harry Furstenberg, Poincaré recurrence and number theory, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 3, 211 – 234. · Zbl 0481.28013 [24] K. de Leeuw and I. Glicksberg, Almost periodic functions on semigroups, Acta Math. 105 (1961), 99 – 140. · Zbl 0104.05601 [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-Berlin, 1979. · Zbl 0439.43001 [26] Miguel de Guzmán, Real variable methods in Fourier analysis, North-Holland Mathematics Studies, vol. 46, North-Holland Publishing Co., Amsterdam-New York, 1981. Notas de Matemática [Mathematical Notes], 75. [27] Paul R. Halmos and John von Neumann, Operator methods in classical mechanics. II, Ann. of Math. (2) 43 (1942), 332 – 350. · Zbl 0063.01888 [28] Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea, Abstract ergodic theorems, Trans. Amer. Math. Soc. 107 (1963), 107 – 124. · Zbl 0119.32701 [29] Ulrich Krengel, On the individual ergodic theorem for subsequences, Ann. Math. Statist. 42 (1971), 1091 – 1095. · Zbl 0216.09603 [30] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. · Zbl 0281.10001 [31] Viktor Losert, A class of sequences with a strong average property, Adv. in Math. 55 (1985), no. 3, 217 – 223. · Zbl 0585.60016 [32] D. Newton and W. Parry, On a factor automorphism of a normal dynamical system, Ann. Math. Statist 37 (1966), 1528 – 1533. · Zbl 0178.52703 [33] John C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116 – 136. · Zbl 0046.11504 [34] Jakob I. Reich, On the individual ergodic theorem for subsequences, Ann. Probability 5 (1977), no. 6, 1039 – 1046. · Zbl 0372.60054 [35] V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation 1952 (1952), no. 71, 55. [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. · Zbl 0324.28009 [37] W. A. Veech, Commentary on $$[{\mathbf{41a}},{\mathbf{b}}]$$, 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-Berlin, 1982. · Zbl 0475.28009 [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. · Zbl 0656.42001 [40] Norbert Wiener and Aurel Wintner, Harmonic analysis and ergodic theory, Amer. J. Math. 63 (1941), 415 – 426. · Zbl 0025.06504 [41] Norbert Wiener and Aurel Wintner, On the ergodic dynamics of almost periodic systems, Amer. J. Math. 63 (1941), 794 – 824. · Zbl 0026.13102 [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. · Zbl 0575.28009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.