Open Access

On strong uniform distribution IV

Journal of Inequalities and Applications20052005:639193

DOI: 10.1155/JIA.2005.319

Received: 24 January 2003

Published: 11 July 2005

Abstract

Let https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq1_HTML.gif be a strictly increasing sequence of natural numbers and let https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq2_HTML.gif be a space of Lebesgue measurable functions defined on https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq3_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq4_HTML.gif denote the fractional part of the real number https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq5_HTML.gif . We say that https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq6_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq7_HTML.gif sequence if for each https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq8_HTML.gif we set https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq9_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq10_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq11_HTML.gif , almost everywhere with respect to Lebesgue measure. Let https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq12_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq13_HTML.gif . In this paper, we show that if https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq14_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq15_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq16_HTML.gif , then there exists https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq17_HTML.gif such that if https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq18_HTML.gif denotes https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq19_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq20_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq21_HTML.gif . We also show that for any https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq22_HTML.gif sequence https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq23_HTML.gif and any nonconstant integrable function https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq24_HTML.gif on the interval https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq25_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.319/MediaObjects/13660_2003_Article_1540_IEq26_HTML.gif , almost everywhere with respect to Lebesgue measure.

Authors’ Affiliations

(1)
Department of Mathematical Sciences, The University of Liverpool

Copyright

© Nair 2005