Hostname: page-component-6bf8c574d5-vmclg Total loading time: 0 Render date: 2025-02-21T10:42:00.045Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Pointwise Convergence Results in Lp(μ), 1 < p < ∞

Published online by Cambridge University Press:  20 November 2018

Richard Duncan
Richard Duncan*
Affiliation:
Department of Mathematics, University of MontrealP.O. Box 6128, MontrealQue. H3C 3J7
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The theory of almost everywhere convergence has its roots in the poineering work of A. Kolmogorov, and today it constitutes one of the most captivating and challenging chapters in modern probability theory and analysis. Whereas some modes of convergence for sequences of measurable functions, e.g. convergence in norm, can be readily obtained by an intelligent exploitation of the various properties of the function spaces involved, a.e. convergence invariably requires a rather high, and sometimes surprising, degree of mathematical virtuosity.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 20 , Issue 3 , 01 September 1977 , pp. 277 - 284
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Akcoglu, M. A. and Sucheston, L., Remarks on dilations in Lp-spaces, Proc. Amer. Math. Soc. 53, 80-82 (1975).Google Scholar
2. Akcoglu, M. A.. A pointwise ergodic theorem in Lp-spaces, Canad. J. Math. 27, 1075-1082 (1975).Google Scholar
3. Akcoglu, M. A. and Miller, H. D. B., Dominated estimates in Hilbert space, Proc. Amer. Math. Soc, 55, 371-271 (1969).Google Scholar
4. Burkholder, D. L., Semi-Gaussian subspaces, Trans. Amer. Math. Soc. 104, 123-131 (1962).Google Scholar
5. Chacon, R. V., A class of linear transformations, Proc. Amer. Math. Soc. 15, 560-564 (1964).Google Scholar
6. Chacon, R. V., and Olsen, J., Dominated estimates of positive contractions, Proc. Amer. Math. Soc. 20, 266-271 (1969).Google Scholar
7. de la Torre, A., A simple proof of the maximal ergodic theorem, Can. J. of Math., vol 28, 1073-75 (1976).Google Scholar
8. Duncan, R. D., Almost everywhere convergence of a class of integrable functions, Z. Wahrscheinliehkeitstheorie verw. Geb. 31, 89-94 (1975).Google Scholar
9. Duncan, R. D., Pointwise convergence theorems for self-adjoint and unitary contractions, Ann. of Probability vol. 5, No. 4, 622-626 (1977).Google Scholar
10. Dunford, N., and Schwartz, J., Linear Operators, Vol. I, Wiley-Interscience, New York, 1958.Google Scholar
11. Garsia, A., Topics in Almost Everywhere Convergence, Chicago, Markham (1970).Google Scholar
12. Hopf, E., The general temporally discrete Markov process, J. Rat. Mech. Anal. 3, 13-45 (1954).Google Scholar
13. Ionescu Tulcea, A., Ergodic properties of isometries in Lp spaces, 1 < p < a ∞, Bull. Amer. Math. Soc, 70, 366-371 (1964).Google Scholar
14. Kaczmarz, S., Sur la convergence et la sommabilité des développements orthogonaux, Studia Math. 1, 87-121 (1929).Google Scholar
15. Lamperti, J., On the isometries of certain function spaces, Pacific J. Math. 8, 459-466 (1958).Google Scholar
16. Lessard, S., Th?se de doctorat, l'université de Montréal.Google Scholar
17. Lorch, E.R., Means of iterated transformations in reflexive vector spaces, Bull. Amer. Math. Soc. 45, 945-947 (1939).Google Scholar
18. Menshov, D., Sur les séries des fonctions orthogonales, Fundamenta Math. 4, 82-105 (1923).Google Scholar
19. Riesz, F., Sur la théorie ergodique, Comm. Math. Helv. 17, 221-239 (1945).Google Scholar
20. Stein, E. M., Topics in harmonie analysis, Annals of Mathematics Studies, no. 63, Princeton University Press (1970).Google Scholar
21. Yosida, K., Functional Analysis, Springer-Verlag, New York (1966).Google Scholar