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Abstrakty
We show that if a = (an)n∈N is a good weight for the dominated weighted ergodic theorem in Lp, p > 1, then the Nörlund matrix Na = {ai−j / Ai}0≤j≤i, Ai = ∑ik=0|ak|, is bounded on ℓp(N). We study the regularity (convergence in norm and almost everywhere) of operators in ergodic theory: power series of Hilbert contractions and power series ∑n∈N an Pn f of L2-contractions, and establish similar close relations to the Nörlund operator associated to the modulus coefficient sequence (|an|)n∈N.
Słowa kluczowe
Wydawca
Rocznik
Tom
Strony
69--85
Opis fizyczny
Bibliogr. 30 poz.
Twórcy
autor
- Institut de Sciences Exactes et Appliquées, Université de la Nouvelle Calédonie, B.P. 4477, F-98847 Noumea Cedex
autor
- IRMA, 10 rue du Général Zimmer, 67084 Strasbourg Cedex, France
Bibliografia
- [1] G. Alexits, Convergence Problems of Orthogonal Series, Int. Ser. Monogr. Pure Appl. Math. 20, Pergamon Press, New York, 1961.
- [2] C. Arhancet and C. Le Merdy, Dilation of Ritt operators on Lp-spaces, Israel J. Math. 201 (2014), 373-414.
- [3] A. Bellow and V. Losert, The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences, Trans. Amer. Math. Soc. 288 (1985), 307-345.
- [4] G. Bennett, Inequalities complimentary to Hardy, Quart. J. Math. 49 (1998), 395-432.
- [5] D. Borwein, Nörlund operators on lp, Canad. Math. Bull. 36 (1993), 8-14.
- [6] D. Borwein and F. P. Cass, Nörlund matrices as bounded operators on lp, Arch. Math. (Basel) 42 (1984), 464-469.
- [7] J. Bourgain, An approach to pointwise ergodic theorems, in: Geometric Aspects of Functional Analysis (1986/87), Lecture Notes in Math. 1317, Springer, Berlin, 1988, 204-223.
- [8] J. Bourgain, Temps de retour pour les systèmes dynamiques, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 483-485.
- [9] J. Bourgain, Pointwise ergodic theorems for arithmetic sets (with an appendix by the author, H. Furstenberg, Y. Katznelson and D. S. Ornstein), Inst. Hautes Études Sci. Publ. Math. 69 (1989), 5-45.
- [10] G. Cohen, C. Cuny and M. Lin, Almost everywhere convergence of powers of some positive Lp contractions, J. Math. Anal. Appl. 420 (2014), 1129-1153.
- [11] G. Cohen, C. Cuny and M. Lin, On convergence of power series of Lp contractions, in: Études opératorielles, Banach Center Publ. 112, Inst. Math., Polish Acad. Sci., Warszawa, 2017, 53-86.
- [12] G. Cohen and M. Lin, Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory, Israel J. Math. 148 (2005), 41-86.
- [13] C. Cuny, Pointwise ergodic theorems with rate with applications to limit theorems for stationary processes, Stoch. Dynam. 11 (2011), 135-155.
- [14] C. Cuny, Norm convergence of some power series of operators in Lp with applications in ergodic theory, Studia Math. 200 (2010), 1-29.
- [15] C. Cuny, Almost everywhere convergence of generalized ergodic transforms for invertible power-bounded operators in Lp, Colloq. Math. 124 (2011), 61-77.
- [16] C. Cuny and M. Lin, Limit theorems for Markov chains by the symmetrization method, J. Math. Anal. Appl. 434 (2016), 52-83.
- [17] C. Cuny and M. Weber, Ergodic theorems with arithmetical weights, Israel J. Math. 217 (2017), 139-180.
- [18] B. Delyon and F. Delyon, Generalization of von Neumann’s spectral sets and integral representation of operators, Bull. Soc. Math. France 127 (1999), 25-41.
- [19] C. Demeter, Improved range in the return times theorem, Canad. Math. Bull. 55 (2012), 708-722.
- [20] C. Demeter, M. Lacey, T. Tao and C. Thiele, Breaking the duality in the return Times theorem, Duke Math. J. 143 (2008), 281-355.
- [21] Y. Derriennic and M. Lin, Fractional Poisson equations and ergodic theorems for fractional coboundaries, Israel J. Math. 123 (2001), 93-130.
- [22] V. F. Gaposhkin, Criteria for the strong law of large numbers for classes of stationary processes and homogeneous random fields, Dokl. Akad. Nauk SSSR 223 (1975), 1044-1047 (in Russian).
- [23] V. F. Gaposhkin, Spectral criteria for existence of generalized ergodic transforms, Theory Probab. Appl. 41 (1996), 247-264; translation from Teor. Veroyatn. Primen. 41 (1996), 251-271.
- [24] C. Le Merdy, H∞ functional calculus and square function estimates for Ritt operators, Rev. Mat. Iberoamer. 30 (2014), 1149-1190.
- [25] B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, and Akadémiai Kiadó, Budapest, 1970.
- [26] J. J. Schäffer, On unitary dilations of contractions, Proc. Amer. Math. Soc. 6 (1955), 322.
- [27] M. Peligrad and S. Utev, A new maximal inequality and invariance principle for stationary sequences, Ann. Probab. 33 (2005), 798-815.
- [28] D. Volný, Martingale-coboundary representation for stationary random fields, Stoch. Dynam. 18 (2018), 1850011, 18 pp.
- [29] M. Weber, Dynamical Systems and Processes, IRMA Lectures in Math. Theoret. Phys. 14, Eur. Math. Soc., Zürich, 2009.
- [30] M. Wierdl, Pointwise ergodic theorem along the prime numbers, Israel J. Math. 64 (1988), 315-336.
Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).
Typ dokumentu
Bibliografia
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