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A Quantitative Version of a de Bruijn‐Post Theorem
Author(s) -
Salvati Simonetta,
Volčič Aljoša
Publication year - 2001
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200109)229:1<161::aid-mana161>3.0.co;2-0
Subject(s) - de bruijn sequence , mathematics , integrable system , sequence (biology) , limit (mathematics) , locally integrable function , riemann hypothesis , function (biology) , pure mathematics , discrete mathematics , mathematical analysis , genetics , evolutionary biology , biology
A theorem due to de Bruijn and Post states that if a real valued function f defined on [0, 1] is not Riemann‐integrable, then there exists a uniformly distributed sequence { x i } such that the averages $\textstyle {1 \over n} \sum\nolimits_{i=1}^n f(x_i)$ do not admit a limit. In this paper we will prove a quantitative version of this result and we will extend it to functions with values in ℝ d .