Borel matrix

Michel Weber

Commentationes Mathematicae Universitatis Carolinae (1995)

  • Volume: 36, Issue: 2, page 401-415
  • ISSN: 0010-2628

Abstract

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We study the Borel summation method. We obtain a general sufficient condition for a given matrix to have the Borel property. We deduce as corollaries, earlier results obtained by G. M“uller and J.D. Hill. Our result is expressed in terms belonging to the theory of Gaussian processes. We show that this result cannot be extended to the study of the Borel summation method on arbitrary dynamical systems. However, in the -setting, we establish necessary conditions of the same kind by using Bourgain’s entropy criterion.

How to cite

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Weber, Michel. "Borel matrix." Commentationes Mathematicae Universitatis Carolinae 36.2 (1995): 401-415. <http://eudml.org/doc/247725>.

@article{Weber1995,
abstract = {We study the Borel summation method. We obtain a general sufficient condition for a given matrix $A$ to have the Borel property. We deduce as corollaries, earlier results obtained by G. M“uller and J.D. Hill. Our result is expressed in terms belonging to the theory of Gaussian processes. We show that this result cannot be extended to the study of the Borel summation method on arbitrary dynamical systems. However, in the $L^p$-setting, we establish necessary conditions of the same kind by using Bourgain’s entropy criterion.},
author = {Weber, Michel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Borel matrix; almost sure convergence; GB and GC sets; Gaussian processes; ergodic theorem; Borel matrix; Borel summation; Bourgain's entropy criterion},
language = {eng},
number = {2},
pages = {401-415},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Borel matrix},
url = {http://eudml.org/doc/247725},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Weber, Michel
TI - Borel matrix
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 2
SP - 401
EP - 415
AB - We study the Borel summation method. We obtain a general sufficient condition for a given matrix $A$ to have the Borel property. We deduce as corollaries, earlier results obtained by G. M“uller and J.D. Hill. Our result is expressed in terms belonging to the theory of Gaussian processes. We show that this result cannot be extended to the study of the Borel summation method on arbitrary dynamical systems. However, in the $L^p$-setting, we establish necessary conditions of the same kind by using Bourgain’s entropy criterion.
LA - eng
KW - Borel matrix; almost sure convergence; GB and GC sets; Gaussian processes; ergodic theorem; Borel matrix; Borel summation; Bourgain's entropy criterion
UR - http://eudml.org/doc/247725
ER -

References

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  1. Beck J., Chen W., Irregularities of distribution, Cambridge Univ. Press, 1987. Zbl1156.11029MR0903025
  2. Beeckmann W., Zeller K., Theorie der Limitierungsverfahren 2, Aufl. Erg. Math. Grenzeb., Springer Verlag, 1970. MR0118990
  3. Bellow A., Losert V., On sequences of density zero in ergodic theory, Proc. Kakutani Conf., 1984. Zbl0587.28013MR0737387
  4. Bourgain J., Almost sure convergence and bounded entropy, Israel J. Math. 63 (1988), 79-95. (1988) Zbl0677.60042MR0959049
  5. Conze J.P., Convergence des moyennes ergodiques pour des sous-suites, Bull. Soc. Math. France 35 (1973), 7-15. (1973) Zbl0285.28017MR0453975
  6. Cooke R.G., Infinite matrices and sequence spaces, Macmillan, London, 1950. Zbl0132.28901MR0040451
  7. Del Junco A., Rosenblatt J., Counterexamples in Ergodic Theory and Number Theory, Math. Ann. 245 (1979), 185-197. (1979) Zbl0398.28021MR0553340
  8. Dudley R.M., The size of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Analysis 1 (1967), 290-330. (1967) MR0220340
  9. Grillenberger C., Krengel U., On matrix summation and the pointwise ergodic theorem, Lecture Notes in Math., Springer 532 (1976), 113-124. (1976) Zbl0331.28010MR0486411
  10. Garsia A., Rodemich E. and Rumsey H. Jr., A real variable lemma and continuity of paths of some Gaussian processes, Indiana Univ. Math. (1970), 565-578. (1970) 
  11. Hill J.D., Remarks on the Borel property, Pacific J. Math. 4 (1954), 227-242. (1954) Zbl0057.29301MR0062244
  12. Krengel U., Ergodic Theorems, W. de Gruyter, 1989. Zbl0649.47042MR0797411
  13. Kuipers L., Niederreiter H., Uniform Distribution of Sequences, J. Wiley Ed., 1971. Zbl0568.10001MR0419394
  14. Müller G., Sätze über Folgen auf kompakten Raümen, Monatsheft. Math. 67 (1963), 436-451. (1963) MR0158199
  15. Peyerimhoff A., Lectures on Summability, Lecture Notes in Math., Springer 107 (1969). (1969) Zbl0182.08401MR0463744
  16. Philipp W., Über einen Satz von Davenport-Erdös-Le Veque, Monatsheft Math. 68 (1964), 52-58. (1964) MR0162784
  17. Schneider D., Weber M., Une remarque sur un Théorème de Bourgain, Séminaire de Probabilités XXVII Lectures Notes in Math., Springer 1557 (1993), 202-206. (1993) Zbl0799.60035MR1308565
  18. Talagrand M., Regularity of Gaussian processes, Acta. Math. 159 (1987), 99-149. (1987) Zbl0712.60044MR0906527
  19. Weber M., Une version fonctionnelle du Théorème ergodique ponctuel, Comptes Rendus Acad. Sci. Paris, Sér. I 311 (1990), 131-133. (1990) Zbl0739.28007MR1065444
  20. Weber M., Méthodes de sommation matricielles, Comptes Rendus Acad. Sci. Paris, Sér. I 315 (1992), 759-764. (1992) Zbl0768.40002MR1184897
  21. Weber M., GC sets, Stein's elements and matrix summation methods, Prépublication IRMA no 027, Université de Strasbourg, 1993. 
  22. Weber M., GB and GC sets in ergodic theory, IXth Conference on Probability in Banach Spaces, Sandberg, August 1993, Denmark, Progress in Prob. V, Birkhauser, t. 35, 1994. Zbl0808.28011MR1308514
  23. Weber M., Coupling of the GB set property for ergodic averages, to appear in J. Theoretic. Prob. (1995), 1993. Zbl0851.60037MR1371072

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