×

Almost everywhere convergence of series. (English) Zbl 0618.60029

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.

MSC:

60F15 Strong limit theorems
60G50 Sums of independent random variables; random walks
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Del Junco, A., Rosenblatt, J.: Counterexamples in ergodic theory and number theory. Math. Ann.245, 185-197 (1979) · Zbl 0408.28015 · doi:10.1007/BF01673506
[2] Dunford, N., Schwartz, J.: Linear operators, Vol. 1. New York; Interscience 1958 · Zbl 0084.10402
[3] Hardy, G.H.: A theorem concerning Taylor series. Q. J. Math. Oxf. II. Ser.44, 147-160 (1913) · JFM 44.0476.04
[4] Kahane, J.P.: Some random series of functions. Lexington: Heath 1968 · Zbl 0192.53801
[5] Kuipers, L., Niederreiter, H.: Uniform distribution of sequences, New York: Wiley 1974 · Zbl 0281.10001
[6] Marcus, M., Pisier, G.: Random Fourier series with applications to harmonic analysis. Ann. Math. Studies 101, Princeton: Princeton University Press 1981 · Zbl 0474.43004
[7] Rosenblatt, J.: Convergence of series of translations. Math. Ann.230, 245-272 (1977) · Zbl 0355.40002 · doi:10.1007/BF01367579
[8] Salem, R., Zygmund, A.: Some properties of trigonometric series whose terms have random signs. Acta Math.91, 245-301 (1954) · Zbl 0056.29001 · doi:10.1007/BF02393433
[9] Sz-Nagy, B., Foias, C.: Analyse harmonique des opérateurs de l’espace de Hilbert. Akad. Kiadó, Budapest. Paris: Mason 1967; English rev. transl. Amsterdam New York: Elsevier/North Holland 1970
[10] Zygmund, A.: Trigonometric series, Vols. 1 and 2. Cambridge: Cambridge University Press 1959 · Zbl 0085.05601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.