Polynomials of best approximation in C[-1,1]
Equivalences between the condition \(| P_ n^{(k)}(x)| \leq K(n^{-1}\sqrt{1-x^ 2}+1/n^ 2)^ kn^{-\alpha},\) where \(P_ n(x)\) is the best n-th degree polynomial approximation to f(x), and the Peetre interpolation space between C[-1,1] and the space \((1-x^ 2)^ kf^{(2k)}(x)\in C[-1,1]\) is established. A similar result is shown for \(E_ n(f)=\inf_{p_ n}\| f-P_ n\|_{C[-1,1]}.\) Rates other than \(n^{-\alpha}\) are also discussed.
scientific article; zbMATH DE number 3931654