Abstract
In the present paper we study the asymptotic behavior of trigonometric products of the form prod _{k=1}^N 2 sin (pi x_k) for N rightarrow infty , where the numbers omega =(x_k)_{k=1}^N 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 omega , 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 omega are the initial segment of a Kronecker or van der Corput sequences The paper concludes with some probabilistic analogues.
Highlights
Introduction and statement of the resultsLet f be a function f : [0, 1] → R+0 andk≥1 be a sequence of numbers in the unit interval
Much work was done on analyzing so-called Weyl sums of the form
It is the aim of this paper to propagate the analysis of corresponding “Weyl products”
Summary
Let f be a function f : [0, 1] → R+0 and (xk)k≥1 be a sequence of numbers in the unit interval. 2 sin(π {nα}) = |2 sin(π nα)| , n=1 n=1 where α is a given irrational number, i.e., we consider the special case when (xn)n≥1 is the Kronecker sequence ({nα})n≥1. Such products play an essential role in many fields and are the best studied such Weyl products in the literature. It is interesting that the constant π/ 6 in Theorem 1 is exactly the same as in results obtained by Fukuyama [13] for products |2 sin(π nkα)| and |2 cos(π nkα)| under the “super-lacunary” gap condition nk+1/nk → ∞. 2 we will prove Theorems 1 and 2, which give estimates of Weyl products in terms of the discrepancy of the numbers (xk)1≤k≤N.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
