On best quadrature formulas for evaluating curvilinear integrals for some classes of functions and curves (Q2343903)

The problem is to find the optimal quadrature formula in the sense of Nikol'skii for integrals of \(f\) over a curve \(\Gamma=\{\mathbf{x}(t): 0\leq t\leq L\}\subset\mathbb{R}^m\). It is shown that if on \(\Gamma\), \(f\) satisfies \(|f(\mathbf{x})-f(\mathbf{y})|\leq \|\mathbf{x}-\mathbf{y}\|_p\), with \(p\) one of \(1,2,\infty\) and if the curve \(\Gamma\) has modulus of continuity \(\boldsymbol{\omega}=(\omega_1,\dots,\omega_m)\), then the midpoint rule for an equidistant partitioning of \([0,L]\) is optimal. The vector \(\boldsymbol{\omega}\) refers to the fact that \(\omega_i\) is the modulus of continuity for the \(i\)th component of \(\mathbf{x}\).