Mean oscillation modulus and number-theoretic grid quadrature formulas

Resumo

For arbitrary Riemann integrable functions f and irrational numbers θ ∈ (0, 1), we obtain estimates of the error R_n(f, θ) of the quadrature formula

\(\int_0^1 {f\left( x \right)dx = \frac{1}{n}\sum\limits_{k = 1}^n {f\left( {\left\{ {k\theta } \right\}} \right) - {R_n}\left( {f,\theta } \right)} } \)

in which {kθ} is the fractional part of the number kθ.