On Weyl products and uniform distribution modulo one
| Autor: | Sumaia Saad Eddin, Friedrich Pillichshammer, Robert F. Tichy, Christoph Aistleitner, Gerhard Larcher |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Uniform distribution (continuous)
General Mathematics Modulo 01 natural sciences Omega Article Combinatorics 010104 statistics & probability symbols.namesake Star-discrepancy Van der Corput sequence Kronecker delta FOS: Mathematics 11K31 Number Theory (math.NT) 0101 mathematics Mathematics 11L15 Mathematics - Number Theory 11K06 11K31 11L15 010102 general mathematics Kronecker sequence 11K06 van der Corput sequence Number theory symbols Trigonometry Trigonometric product Unit interval |
| Zdroj: | Monatshefte Fur Mathematik |
| Popis: | In the present paper we study the asymptotic behavior of trigonometric products of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\prod _{k=1}^N 2 \sin (\pi x_k)$$\end{document}∏k=1N2sin(πxk) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \rightarrow \infty $$\end{document}N→∞, where the numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega =(x_k)_{k=1}^N$$\end{document}ω=(xk)k=1N are evenly distributed in the unit interval [0, 1]. The main result are matching lower and upper bounds for such products in terms of the star-discrepancy of the underlying points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}ω, thereby improving earlier results obtained by Hlawka (Number theory and analysis (Papers in Honor of Edmund Landau, Plenum, New York), 97–118, 1969). Furthermore, we consider the special cases when the points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}ω are the initial segment of a Kronecker or van der Corput sequences The paper concludes with some probabilistic analogues. |
| Databáze: | OpenAIRE |
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