Uniform estimates of monotone and convex approximation of smooth functions
The paper estimates the degree of approximation of monotone and of convex smooth functions by means of monotone and convex polynomials, respectively. A function \(f\in C[-1, 1]\cap C^ r(- 1,1)\) is said to be in the class \(B^ r\), if \(\sup_{0< x< 1} |(1- x^ 2)^{r/2} f^{(r)}(x)|< \infty\). Let \(\varphi(x):= \sqrt{1- x^ 2}\) and define for \(k\geq 1\), the weighted Ditzian-Totik modulus of smoothness of \(f\) by \[ \omega^ k_{\varphi, r}(f, t):= \sup_{0< h\leq r} \|(1- kh\varphi(x)/2- x)^{r/2} (1- kh\varphi(x)/2- x)^{r/2} \Delta^ k_{h\varphi(z)}(f, \cdot)\|, \] where \(\Delta^ k_{h\varphi(x)}\) is the familiar non-uniform difference. Denote by \(E^{(1)}_ n (f)\), and \(E^{(2)}_ n (f)\), the degrees of monotone and convex approximation by polynomials, respectively. Then the paper is devoted into proving the following significant results: Theorem 1. For \(k\geq 1\) and \(r\geq 3\), there exists a constant \(C= C(r, k)\), such that if \(f\in B^ r\) is nondecreasing, then for every \(n\geq r+ k-1\), \[ E^{(1)}_ n (f)\leq Cn^{-r} \omega^ k_{\varphi, r}(f^{(r)}, 1/n). \] Theorem 2. For \(k\geq 1\) and \(r\geq 5\), there exists a constant \(C= C(r, k)\), such that if \(f\in B^ r\) is convex, then for every \(n\geq r+ k- 1\), \[ E^{(2)}_ n (f)\leq Cn^{-r} \omega^ k_{\varphi, r} (f^{(r)}, 1/n). \] It is known that when \(r< 3\), in the monotone case, and when \(r< 5\) in the convex case, estimates like the above are usually impossible to obtain. In fact in the monotone case they hold only when \(r+ k\leq 2\), and in the convex case only when \(r+ k\leq 3\).
scientific article; zbMATH DE number 720667