Ergodic averaging sequences


We consider generalizations of the pointwise and mean ergodic theorems to ergodic theorems averaging along different subsequences of the integers or real numbers. The Birkhoff and Von Neumann ergodic theorems give conclusions about convergence of average measurements of systems when the measurements are made at integer times. We consider the case when the measurements are made at times a(n) or (⌊a(n)⌋) where the function a(x) is taken from a class of functions called a Hardy field, and we also assume that |a(x)| goes to infinity more slowly than some positive power of x. A special, well-known Hardy field is Hardy's class of logarithmico-exponential functions. The main theme of the paper is to point out that for a function a(x) as described above, a complete characterization of the ergodic averaging behavior of the sequence (⌊a(n)⌋) is possible in terms of the distance of a(x) from (certain) polynomials.

Publication Title

Journal d'Analyse Mathematique