TY - JOUR

T1 - Extensions of the Menchoff-Rademacher theorem with applications to Ergodic theory

AU - Cohen, Guy

AU - Lin, Michael

N1 - Funding Information:
ACKNOWLEDGEMENTS: 1. The first author was partially supported by grant 235/01 of the Israel Science Foundation.

PY - 2005/12/1

Y1 - 2005/12/1

N2 - We prove extensions of Menchoff's inequality and the MenchoffRademacher theorem for sequences {fn} ⊂ Lp, based on the size of the norms of sums of sub-blocks of the first n functions. The results are applied to the study of a.e. convergence of series ∑n a nTng/nα when T is an L 2-contraction, g ε L2, and {an} is an appropriate sequence. Given a sequence {fn} sub; L P(Ω,μ), 1 < p ≤ 2, of independent centered random variables, we study conditions for the existence of a set of x of μ-probability 1, such that for every contraction T on L 2(γ,π) and g ε L2(π), the random power series ∑n fn(x)Tng converges π-a.e. The conditions are used to show that for {fn} centered i.i.d. with f 1 ε L log+ L, there exists a set of x of full measure such that for every contraction T on L2(γ, π) and g ε L2(π), the random series ∑n fn(x)T np/n converges π-a.e.

AB - We prove extensions of Menchoff's inequality and the MenchoffRademacher theorem for sequences {fn} ⊂ Lp, based on the size of the norms of sums of sub-blocks of the first n functions. The results are applied to the study of a.e. convergence of series ∑n a nTng/nα when T is an L 2-contraction, g ε L2, and {an} is an appropriate sequence. Given a sequence {fn} sub; L P(Ω,μ), 1 < p ≤ 2, of independent centered random variables, we study conditions for the existence of a set of x of μ-probability 1, such that for every contraction T on L 2(γ,π) and g ε L2(π), the random power series ∑n fn(x)Tng converges π-a.e. The conditions are used to show that for {fn} centered i.i.d. with f 1 ε L log+ L, there exists a set of x of full measure such that for every contraction T on L2(γ, π) and g ε L2(π), the random series ∑n fn(x)T np/n converges π-a.e.

UR - http://www.scopus.com/inward/record.url?scp=32544447715&partnerID=8YFLogxK

U2 - 10.1007/BF02775432

DO - 10.1007/BF02775432

M3 - Article

AN - SCOPUS:32544447715

VL - 148

SP - 41

EP - 86

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

ER -