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2 Bulletin of the Polish Academy of Sciences. Mathematics

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On Nörlund summation and ergodic theory, with applications to power series of Hilbert contractions

Cuny C., Weber M.

Bulletin of the Polish Academy of Sciences. Mathematics

2018

Vol. 66, no. 1

69--85

We show that if a = (an)n∈N is a good weight for the dominated weighted ergodic theorem in Lp, p > 1, then the Nörlund matrix Na = {ai−j / Ai}0≤j≤i, Ai = ∑ik=0|ak|, is bounded on ℓp(N). We study the regularity (convergence in norm and almost everywhere) of operators in ergodic theory: power series of Hilbert contractions and power series ∑n∈N an Pn f of L2-contractions, and establish similar close relations to the Nörlund operator associated to the modulus coefficient sequence (|an|)n∈N.

On Billard's theorem for random Fourier series

Cohen G., Cuny C.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 1

39-53

We show that Billard's theorem on a.s. uniform convergence of random Fourier series with independent symmetric coefficients is not true when the coefficients are only assumed to be centered independent. We give some necessary or sufficient conditions to ensure the validity of Billard's theorem in the centered case.