Degree of simultaneous coconvex polynomial approximation
Let \(f\in C^1[-1,1]\) change its convexity \(s\)-times at the points \(y_j\in (-1,1)\) \((j-1, \dots,s)\). Then \(f\) is approximated by polynomials \(p_n\), which are coconvex with \(f\), i.e., \(p_n\) changes its convexity exactly at the same points \(y_j\) \((j=1, \dots,s)\). It is proved that for each sufficiently large \(n\), there exists a polynomial \(p_n\) of degree \(n\), which is coconvex with \(f\) and satisfies \[ \begin{aligned} \| f-p_n \|_\infty & \leq C(s)n^{-1} \omega_2 (f^1, n^{-1}),\\ \| f'-p_n' \|_\infty & \leq C(s) \omega_2 (f',n^{-1}). \end{aligned} \] The constant \(C(s)\) is independent of \(f\), \(n\), and the location of the points \(y_j\) \((j=1, \dots, s)\). Here \(\omega_2\) denotes the second Ditzian-Totik modulus of smoothness.
scientific article; zbMATH DE number 1209937