Israel Journal of Mathematics

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

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



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.


Spectral Density Maximal Function Orthogonal Series Pointwise Ergodic Theorem Dual Index 
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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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