scholarly journals Almost uniform convergence in the Wiener–Wintner ergodic theorem

2021 ◽  
Author(s):  
Vladimir Chilin ◽  
Semyon Litvinov
Keyword(s):  
Uniform Convergence ◽  
Ergodic Theorem ◽  
Almost Uniform Convergence

scholarly journals Noncommutative strong maximals and almost uniform convergence in several directions – Addendum

2021 ◽  
Vol 9 ◽  
Author(s):  
José M. Conde-Alonso ◽  
Adrián M. González-Pérez ◽  
Javier Parcet
Keyword(s):  
Uniform Convergence ◽  
Almost Uniform Convergence

scholarly journals Uniform Convergence of Cesaro Averages for Uniquely Ergodic C*-Dynamical Systems

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 987 ◽  
Author(s):  
Francesco Fidaleo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Uniform Convergence ◽  
Unit Circle ◽  
Ergodic Theorem ◽  
Ergodic Theory ◽  
C Algebra ◽  
Cesaro Averages ◽  
Uniformly Convergent ◽  
Uniquely Ergodic

Consider a uniquely ergodic C * -dynamical system based on a unital *-endomorphism Φ of a C * -algebra. We prove the uniform convergence of Cesaro averages 1 n ∑ k = 0 n − 1 λ − n Φ ( a ) for all values λ in the unit circle, which are not eigenvalues corresponding to “measurable non-continuous” eigenfunctions. This result generalizes an analogous one, known in commutative ergodic theory, which turns out to be a combination of the Wiener–Wintner theorem and the uniformly convergent ergodic theorem of Krylov and Bogolioubov.

On Almost Uniform Convergence of Families of Functions

1964 ◽  
Vol 7 (1) ◽  
pp. 45-48 ◽  
Author(s):  
Elias Zakon
Keyword(s):  
Uniform Convergence ◽  
Present Note ◽  
Measure Zero ◽  
Slight Modification ◽  
Real Parameter ◽  
General Proof ◽  
Almost Uniform Convergence

In [5] Tolstov showed by a counterexample that Egoroff' s theorem on almost uniform convergence cannot be extended to families of functions (ft(x)}, with t a continuous real parameter. However, Frumkin [2] proved that this is possible provided that some sets of measure zero (depending on t) are disregarded when each particular ft(x) is considered. This interesting result was obtained by using the rather involvec machinery of Kantorovitch' s semi-ordered spaces and Lp spaces. In the present note we intend to give a simpler and more general proof. Indeed, it will be seen that only a slight modification of the standard proof of Egoroff's theorem is necessary to obtain Frumkin' s theorem in a more general form. We shall establish the following result.

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