On monotone rational approximation (Q2567265)

The author solves a problem raised by R. A. DeVore in his several lectures during the last 15 years. The author finds the exact order of monotone rational approximation for the Sobolev classes \(W_p^1\). He proves that if \(f\) is an absolutely continuous function on \([0, 1]\) satisfying \(f'\in L_p[0,1],\) \(1<p<\infty\). If \(f\) is a monotone function on \([0, 1]\) then \[ f_n^{(1)(f)} < \frac{c(p)}{n} \|f'\|_p,\quad n= 1, 2,\dots \] where \(c(p)\) is a constant depending only on \(p\).