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Israel Journal of Mathematics

, Volume 148, Issue 1, pp 41–86 | Cite as

Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory

  • Guy Cohen
  • Michael Lin
Article

Abstract

We prove extensions of Menchoff's inequality and the Menchoff-Rademacher theorem for sequences {f n } ∪L p , based on the size of the norms of sums of sub-blocks of the firstn functions. The results are aplied to the study of a.e. convergence of series Σ n a n T n g/ n whenT is anL 2 -contraction,gL 2 , and {a n } is an appropriate sequence.

Given a sequence {f n }∪L p (Ω, μ), 1<p≤2, of independent centered random variables, we study conditions for the existence of a set ofx of μ-probability 1, such that for every contractionT on\(L_2 (\mathcal{Y},\pi )\) andgL 2 (π), the random power series Σ n f n (x)T n g converges π-a.e. The conditions are used to show that for {f n } centered i.i.d. withf 1L log+ L, there exists a set ofx of full measure such that for every contractionT on\(L_2 (\mathcal{Y},\pi )\) andgL 2 (π), the random series Σ n f n (x)T n g/n converges π-a.e.

Keywords

Spectral Density Maximal Function Orthogonal Series Pointwise Ergodic Theorem Dual Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© The Hebrew University Magnes Press 2005

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceBen-Gurion University of the NegevBeer ShevaIsrael

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