Approximation of a convex function by convex algebraic polynomials in \(L_ p,1{\leq}p<\infty\)
The paper deals with the best approximation of a convex function by convex algebraic polynomials in \(L_ p\), \(1\leq p<\infty\). Pending a number of lemmas it is proved: Let \(f\) be a convex function on \([-1,1]\) such that \(\max_{x\in[-1,1]} f(x)=1\), \(\min_{x\in [-1,1]} f(x)=0\). Then for every \(n\) there exists a convex polynomial \(Q_ n\) of degree at most \(n\) such that \(\| f-Q_ n\|_ p\leq An^{-2/p}\), \(1\leq p<\infty\), where \(A\) is a constant independent of \(f\) and \(p\).
scientific article; zbMATH DE number 205441