Generalization of two results of the theory of uniform distribution
HTML articles powered by AMS MathViewer
- by Petko D. Proĭnov PDF
- Proc. Amer. Math. Soc. 95 (1985), 527-532 Request permission
Abstract:
For a sequence ${x_1}, \ldots ,{x_N}$ of points in $[0,1]$ and a sequence ${p_1}, \ldots ,{p_N}({p_1} + {p_2} + \cdots + {p_N} = 1)$ of nonnegative numbers, define the distribution function \[ g(x) = x - \sum \limits _{{x_k} < x} {{p_k}.} \] Let $\varphi$ be an increasing function on $[0,1]$ and $\varphi (0) = 0$. The main result of the paper is \[ F({D_N}) \leqslant \int _0^1 {\varphi \left ( {\left | {g(x)} \right |} \right )} dx \leqslant \varphi ({D_N}),\] where ${D_N}$ is the supremum norm of $g$ on $[0,1]$ and $F$ is the antiderivative of $\varphi$ with $F(0) = 0$. This result generalizes and improves an estimate of Niederreiter [1] for the ${L^2}$ discrepancy of the sequence ${x_1}, \ldots ,{x_N}$. Applying the above inequality we also obtain a new criterion for uniform distribution modulo one.References
- H. Niederreiter, Application of Diophantine approximations to numerical integration, Diophantine approximation and its applications (Proc. Conf., Washington, D.C., 1972) Academic Press, New York, 1973, pp. 129–199. MR 0357357
- I. M. Tsobol′, Mnogomernye kvadraturnye formuly i funktsii Khaara, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0422968 P. D. Proinov, Note on the convergence of the general quadrature process with positive weights, Constructive Function Theory’77 (Bl. Sendov and D. Vačov, eds.), Sofia, 1980, pp. 121-125.
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 95 (1985), 527-532
- MSC: Primary 11K38
- DOI: https://doi.org/10.1090/S0002-9939-1985-0810157-8
- MathSciNet review: 810157