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Borel matrix. (English) Zbl 0837.28014
The author studies the Borel summation method. Let \(A= \{a_{n,k}, n,k\geq 1\}\) be an infinite matrix with \(\sum^\infty_{k= 1} a_{n, k}= 1\) \((n\geq 1)\) and \(\sup|a_n|_1< \infty\). If \(X\) is compact metrizable and \(\mu\) a Borel probability measure, consider \(\widetilde S_n f= \sum^\infty_{k= 1} a_{n, k} f\circ T\), where \(T\) is the shift on \((X^\infty, \mu^\infty)\). If this converges a.e. to \(\int f d\mu\) for all continuous \(f\), \(A\) is called Borel. Roughly speaking, a sequence \(a_n\) in \(l_1\) is a \(G(1)\)-set if there exist, for any \(\varepsilon> 0\), sequences \((b_n)\) with \(\lim|b_n|_2= 0\), \(\limsup|a_n- b_n|_1\leq \varepsilon\) such that \((b_n)\) is a \(GC\)-set in \(l_2\) (a concept defined by Dudley in terms of Gaussian processes).
For such \(A\), the author proves the convergence \(\mu^\infty\)-a.e. of \(\widetilde S_n f\) for \(f\) in \(L_\infty(\mu^\infty)\) of a probability space, and for other \(f\)’s “generated” from such special \(f\)’s. However, it is shown that the result cannot be extended to arbitrary dynamical systems. In the \(L_p\)-setting, necessary conditions are established using Bourgain’s entropy criterion.

28D05 Measure-preserving transformations
40G10 Abel, Borel and power series methods
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