Approximation by Riemann sums in modular spaces (Q5945550)

Let \(Q=[ 0,1] \) and let \(L^{0}( Q) \) be the space of all Lebesgue measurable functions \(f:Q\to\mathbb{R}.\) A functional \(\rho:L^{0}( Q) \to[ 0,\infty] \) which satisfies the conditions: (i) \(\rho( f) =0\Longleftrightarrow f=0\) a.e. in \(Q\); (ii) \(\rho( -f) =\rho( f) \), for every \(f\in L^{0}( Q) \) and (iii) \(\rho( \alpha f+\beta g) \leq\rho( f) +\rho( g) \), for every \(f,g\in L^{0}( Q) \) and \(\alpha,\beta\geq 0,\alpha+\beta=1,\) is called a modular on \(L^{0}( Q) \), and the space \(L^{\rho}( Q) =\{ f\in L^{0}( Q) :\lim_{\lambda \downarrow 0}\rho( \lambda f) =0\} \) is called the modular space generated by \(\rho\). A sequence \(( f_{n}) \subset L^{\rho }( Q) \) is called \(\rho\) -convergent to \(f\in L^{\rho}( Q) \) if there exists \(\lambda>0\) such that \(\lim_{n\to \infty}\rho( \lambda( f_{n}-f)) =0\). For \(f\in L^{0}( Q) \) and \(y\in Q\), the expression \(R_{n}( f;y) =\frac{1}{n}\sum_{k=0}^{n-1}f( \frac{y+k}{n}) \) is called the translated equidistant Riemann sum for \(f\) and the quantity \(\omega_{\rho}( f,\delta) =\sup_{|s|\leq\delta}\rho[ f( \cdot+s) -f( \cdot) ] \) is called the \(\rho\) -modulus of continuity for \(f\in L^{\rho }( Q) \). The authors obtain estimations for the approximation error of the integral of a function \(f\in L^{\rho}( Q) \) by expressions of the form \(R_{n}( f;y) \) in terms of the \(\rho\) -modulus of continuity. The developed theory is applied to some examples.