Almost everywhere convergence of series
Let (X,\(\beta\),m) be a probability space and let \(T: L_ 2(X)\to L_ 2(X)\) be a contraction. The series \(\sum ^{\infty}_{n=1}c_ nT^ n\) converges in norm if \(\sum ^{\infty}_{n=1}c_ n\exp (2\pi inx)\) converges uniformly. But also there are many examples of \((c_ n)\) for which \(\sum ^{\infty}_{n=1}| c_ n| =\infty\) and yet the series \(\sum ^{\infty}_{n=1}c_ nT^ nf(x)\) converges a.e. x too. Generally, if \(\sum ^{\infty}_{n=1}| c_ n| ^ 2n^{1/2}\log ^ s(n)<\infty\) for some \(s>2\), then for a.e. choice of \(sign(\gamma _ n)\in \{-1,1\}\), if T is any contraction on \(L_ 2(X)\), the series \(S=\sum ^{\infty}_{n=1}\gamma _ nc_ nT^ n\) converges in norm and for all \(f\in L_ 2(X)\), Sf(X) converges a.e. x. By the same technique, one can show that for all \(\sigma >3/4\), and for all \(f\in L_ 2(X)\), the series \(\sum ^{\infty}_{n=1}(\cos (n \log n)/n^{\sigma})T^ nf(x)\) converges a.e. x. By using complex interpolation, similar results can be obtained in \(L_ p(X)\), \(1<p<\infty\). For instance, for a.e. choice of \(sign(\gamma _ n)\in \{- 1,1\}\), if T is a contraction on all \(L_ p(X)\), \(1\leq p\leq \infty\), then for \(1<p<\infty\), \(S=\sum ^{\infty}_{n=1}(\gamma _ n/n)T^ n\) converges in norm, and for all \(f\in L_ p(X)\), Sf(x) converges a.e. x. General theorems of these types, other examples and analogies, and some applications are discussed.
scientific article; zbMATH DE number 4003154