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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.