On strong uniform distribution IV
<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>...
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2005-01-01
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| Series: | Journal of Inequalities and Applications |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/2005/639193 |
| _version_ | 1818307988043071488 |
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| author | Nair R |
| author_facet | Nair R |
| author_sort | Nair R |
| collection | DOAJ |
| description | <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> |
| first_indexed | 2024-12-13T07:07:07Z |
| format | Article |
| id | doaj.art-ae345c2f59cf4fb68ba9e4b5fb0bcae8 |
| institution | Directory Open Access Journal |
| issn | 1025-5834 1029-242X |
| language | English |
| last_indexed | 2024-12-13T07:07:07Z |
| publishDate | 2005-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-ae345c2f59cf4fb68ba9e4b5fb0bcae82022-12-21T23:55:48ZengSpringerOpenJournal of Inequalities and Applications1025-58341029-242X2005-01-0120053639193On strong uniform distribution IVNair R<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>http://www.journalofinequalitiesandapplications.com/content/2005/639193 |
| spellingShingle | Nair R On strong uniform distribution IV Journal of Inequalities and Applications |
| title | On strong uniform distribution IV |
| title_full | On strong uniform distribution IV |
| title_fullStr | On strong uniform distribution IV |
| title_full_unstemmed | On strong uniform distribution IV |
| title_short | On strong uniform distribution IV |
| title_sort | on strong uniform distribution iv |
| url | http://www.journalofinequalitiesandapplications.com/content/2005/639193 |
| work_keys_str_mv | AT nairr onstronguniformdistributioniv |