Abstract
During the two decades after the publication of the pioneering papers “Proof of the ergodic theorem” and “Proof the quasi-ergodic hypothesis” by Birkhoff [4] and von Neumann [18], but before the publication of “A general ergodic theorem” by Calderón in 1953, there were new proofs and generalizations of the results of Birkhoff and von Neumann given by many mathematicians. These include, among others, Khintchine, Hopf, Kolmogorov, F. Riesz, Yoshida, Kakutani, Wiener, Dunford, Pitt, Doob, and Zygmund. Much of this substantial body of work is referenced in [11], [15], and [16]. Explorations in many new directions have continued since then and have greatly expanded the connections and applications of ergodic theory to other parts of mathematics and beyond. Calderón’s 1953 paper (5) played an important role in this development.
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References
Mustafa Akcoglu, Alexandra Bellow, Roger L. Jones, Viktor Losert, Karin Reinhold-Larsson, and Maté Wierdl, The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters, Ergodic Theory Dynam. Systems, 16 (1996), no. 2, 207-253.
Alexandra Bellow, Transference principles in ergodic theory, Harmonic Analysis and Partial Differential Equations, edited by Michael Christ, Carlos E. Kenig, and Cora Sadosky, Univ. Chicago Press, Chicago, Illinois, (1999), 27-39.
Alexandra Bellow and Ulrich Krengel, On Hopfs ergodic theorem for particles with différent velocities, Almost Everywhere Convergence II, edited by Alexandra Bellow and Roger L. Jones, Academic Press, New York, (1991), 41-47.
G. D. Birkhoff, Proof of the ergodic theorem, Proc. Nat. Acad. Sci. U.S.A., (1931), 656-660.
J. Bourgain, Extension of a result of Benedek, Calderón and Panzone, Ark. Mat., 22, (1984), 91-95.
Donald L. Burkholder, Martingales and singular integrals in &Qmach spaces, Handbook of the Geometry of Banach spaces, 1, edited by W. B. Johnson and J. Linden-strauss, Elsevier, Amsterdam, (2001), 233-269.
Ronald R. Coif m an and Guido Weiss, Transference Methods in Analysis, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 31, American Mathematical Society, Providence, R.I., 1976.
Misha Cotlar, A unifed theory of Hubert transforms and ergodic theorems, Rev. Mat. Guyana 1, (1955), 105-167.
Nelson Dunford, An ergodic theorem for noncommutatwe transformations, Acta Sei. Math. Szeged 14, (1951), 1-4.
Eberhard Hopf, Über die Bedeutung der willkürlichen Funktionen für die Wahrscheinlichkeitstheorie, Jahresber. Dtsch. Math. Verein 46, (1936), 179-195.
Eberhard Hopf, Ergodentheone, Springer, Berlin, 1937.
Lars Hörmander, Estimates for translation invariant operators in L p spaces, Acta Math., 104, (1960), 93-139.
Roger L. Jones, Robert Kaufman, Joseph M. Rosenblatt, and Maté Wierdl, Oscillation in ergodic theory, Ergodic Theory Dynam. Systems, 18, (1998), 889-935.
Roger L. Jones, Joseph M. Rosenblatt, Maté Wierdl, Oscillation in ergodic theory: higher dimensional results, Israel J. Math. 135 (2003), 1-27.
Shizuo Kakutani, Ergodic theory, Proc. Int. Congr. Math., 2, (1950), 128-142.
Ulrich Krengel, Ergodic Theorems, De Gruyter, Berlin, 1985.
Elon Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146, (2001), 259-295.
John von Neumann, Proof of the quasi-ergodic hypothesis, Proc. Nat. Acad. Sei. U.S.A., 18, (1932), 70-82.
Amos Nevo, Harmonic analysis and pointwise ergodic theorems for noncommuting transformations, J. Amer. Math. Soc. 7, (1994), 875-902.
Amos Nevo and Elias M. Stein, A generalization of Birkhoff ’s pointwise ergodic theorem, Acta Math., 173, (1994), 135-154.
J. Schwartz, A remark on inequalities of C alder on-Zygmund type for vector-valued functions, Coram. Pure and Applied Math., 14, (1961), 785-799.
A. A. Tempelman, Ergodic theorems for general dynamic systems, Soviet Math. Dokl., 8, (1967), No. 5, 1213-216.
Arkady Tempelman, Ergodie Theorems for Group Actions: Informational and Ther-modynamical Aspects, Kluwer Academic Publishers, Dordrecht, 1992.
Norbert Wiener, The ergodtc theorem, Duke Math. J., 5, (1939), 1-18.
Felipe Zo, A note on approximation of the identity, Studia Math., 55, (1976), 111-122.
A. Zygmund, An individual ergodic theorem for non-commutative transformations, Acta Sei. Math. Szeged 14, (1951), 103-110.
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Davis, B., Song, R. (2011). Comments on (5), (14), (22), (30), and (31). In: Davis, B., Song, R. (eds) Selected Works of Donald L. Burkholder. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7245-3_42
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