On strong uniform distribution IV

Journal of Inequalities and Applications. 2005;2005(3):639193

 

Journal Homepage

Journal Title: Journal of Inequalities and Applications

ISSN: 1029-242X (Online)

Publisher: SpringerOpen

LCC Subject Category: Science: Mathematics

Country of publisher: United Kingdom

Language of fulltext: English

Full-text formats available: PDF, HTML

 

AUTHORS


Nair R

EDITORIAL INFORMATION

Blind peer review

Editorial Board

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Time From Submission to Publication: 13 weeks

 

Abstract | Full Text

<p/> <p>Let <inline-formula><graphic file="1029-242X-2005-639193-i1.gif"/></inline-formula> be a strictly increasing sequence of natural numbers and let <inline-formula><graphic file="1029-242X-2005-639193-i2.gif"/></inline-formula> be a space of Lebesgue measurable functions defined on <inline-formula><graphic file="1029-242X-2005-639193-i3.gif"/></inline-formula>. Let <inline-formula><graphic file="1029-242X-2005-639193-i4.gif"/></inline-formula> denote the fractional part of the real number <inline-formula><graphic file="1029-242X-2005-639193-i5.gif"/></inline-formula>. We say that <inline-formula><graphic file="1029-242X-2005-639193-i6.gif"/></inline-formula> is an <inline-formula><graphic file="1029-242X-2005-639193-i7.gif"/></inline-formula> sequence if for each <inline-formula><graphic file="1029-242X-2005-639193-i8.gif"/></inline-formula> we set <inline-formula><graphic file="1029-242X-2005-639193-i9.gif"/></inline-formula> <inline-formula><graphic file="1029-242X-2005-639193-i10.gif"/></inline-formula>, then <inline-formula><graphic file="1029-242X-2005-639193-i11.gif"/></inline-formula>, almost everywhere with respect to Lebesgue measure. Let <inline-formula><graphic file="1029-242X-2005-639193-i12.gif"/></inline-formula> <inline-formula><graphic file="1029-242X-2005-639193-i13.gif"/></inline-formula>. In this paper, we show that if <inline-formula><graphic file="1029-242X-2005-639193-i14.gif"/></inline-formula> is an <inline-formula><graphic file="1029-242X-2005-639193-i15.gif"/></inline-formula> for <inline-formula><graphic file="1029-242X-2005-639193-i16.gif"/></inline-formula>, then there exists <inline-formula><graphic file="1029-242X-2005-639193-i17.gif"/></inline-formula> such that if <inline-formula><graphic file="1029-242X-2005-639193-i18.gif"/></inline-formula> denotes <inline-formula><graphic file="1029-242X-2005-639193-i19.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2005-639193-i20.gif"/></inline-formula> <inline-formula><graphic file="1029-242X-2005-639193-i21.gif"/></inline-formula>. We also show that for any <inline-formula><graphic file="1029-242X-2005-639193-i22.gif"/></inline-formula> sequence <inline-formula><graphic file="1029-242X-2005-639193-i23.gif"/></inline-formula> and any nonconstant integrable function <inline-formula><graphic file="1029-242X-2005-639193-i24.gif"/></inline-formula> on the interval <inline-formula><graphic file="1029-242X-2005-639193-i25.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2005-639193-i26.gif"/></inline-formula>, almost everywhere with respect to Lebesgue measure.</p>