Pointwise Estimates for $ 3$ -monotone Approximation

Abstract

We prove that for a $ 3$ -monotone function $ F\in C[-1,1]$ , one can achieve the pointwise estimates \[ |F(x)-\Psi(x)|\le c\omega_3(F,\rho_n(x)), \quad x\in[-1,1], \] where $ \rho_n(x):=\frac1{n^2}+\frac{\sqrt{1-x^2}}n$ and $ c$ is an absolute constant, both with $ \Psi$ , a $ 3$ -monotone quadratic spline on the $ n$ th Chebyshev partition, and with $ \Psi$ , a~$ 3$ -monotone polynomial of degree $ \le n$ . The basis for the construction of these splines and polynomials is the construction of $ 3$ -monotone splines, providing appropriate order of pointwise \linebreak approximation, half of which nodes are prescribed and the other half are free, but controlled'\'''\''.