Optimality of piecewise image-linear interpolating operators (Q2444305)

Let \(\tau \in C[0,1]\) be a strictly increasing function such that \(\tau (0) = 0\) and \(\tau (1) = 1\). Consider a partition \(\Delta : 0 = x_0 <x_1 < \dots <x_n =1\) of the interval \([0,1]\), and let \(\tau _i:=\tau (x_i)\). The operator of piecewise \(\tau -\)linear interpolation is defined by \[ L_\Delta ^\tau f(x):= f(x_{i-1}) \frac{\tau (x)-\tau _i}{\tau _{i-1}-\tau _i} + f(x_i)\frac{\tau (x) - \tau _{i-1}}{\tau _i - \tau _{i-1}}, \] \(f\in C[0,1]\), \(x\in [x_{i-1},x_i]\), \(i=1,\dots , n\). It is proved that, in a suitable class of positive linear operators, \(L_\Delta ^\tau\) has a property of optimality with respect to the degree of approximation. Several known results are extended and generalized.