Ergodic Theory and its Connections with Harmonic Analysis: by Karl E. Petersen, Ibrahim Salama

By Karl E. Petersen, Ibrahim Salama

This quantity includes articles that describe the connections among ergodic concept and convergence, stress idea, and the idea of joinings. those papers current the history of every region of interplay, the main extraordinary fresh effects, and the at the moment promising traces of study. within the mixture, they are going to offer an ideal creation for somebody starting examine in a single of those parts.

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Sample text

Let P) be a probability space, and let 0 < o, < 1. Let Y1,Y2... d. sequence of ((-1)-1)-valued random variables wih P(Yn = 1) = o and P(Yn = -1) = 1 -v. Denote an(w) = Ek

A) Show that the sequence A is good for periodic systems by showing that A(q, h) = Ok (q, h) /q, where Bk (q, h) is the number of solutions of the congruence xk - h mod q. b) Show that A(b/q) = 1 E e(nkb/q) = 1 q n

F(t) = oo. a) Show that there is a strictly increasing sequence A of positive integers which is ergodic and satisfies t+00 F(t) = 0. ) b) Is there is a strictly increasing sequence A of positive integers which is ergodic and satisfies A(,) ? 3. Let A = (an) be a strictly increasing sequence of positive integers. Show that A is ergodic if and only if for each 0 E (0, 1) t A(t) 0. 4. a) Let 1 <_ p < oo. Prove that Mt (7G+, f) -* f in LP norm, where f is T-invariant. ) b) Can we take p = oo in part a)?

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