Rates of Uniform Convergence for Riemann Integrals

Convergence rates of approximate sums of Riemann integrals

The Riemann sums and the trapezoidal sums of functions defined on a bounded closed interval are well known as approximate sums of the Riemann integrals of the functions. In this paper the author represents the convergence rates of the Riemann sums and the trapezoidal sums as some limits of their expanded error terms. Let [a, b] be a bounded closed interval. We take an n-division ∆ of [a, b] def...

Integral Inequalities for h(x)-Riemann-Liouville Fractional Integrals

In this article, we obtain generalizations for Grüss type integral inequality by using h(x)-Riemann-Liouville fractional integral.

ON CONVERGENCE THEOREMS FOR FUZZY HENSTOCK INTEGRALS

The main purpose of this paper is to establish different types of convergence theorems for fuzzy Henstock integrable functions, introduced by  Wu and Gong cite{wu:hiff}. In fact, we have proved fuzzy uniform convergence theorem, convergence theorem for fuzzy uniform Henstock integrable functions and fuzzy monotone convergence theorem. Finally, a necessary and sufficient condition under which th...

Uniform Convergence Rates for Kernel Density Estimation

Kernel density estimation (KDE) is a popular nonparametric density estimation method. We (1) derive finite-sample high-probability density estimation bounds for multivariate KDE under mild density assumptions which hold uniformly in x ∈ R and bandwidth matrices. We apply these results to (2) mode, (3) density level set, and (4) class probability estimation and attain optimal rates up to logarit...

Uniform convergence of single and double sine series and integrals