Polynomials with bounds and numerical approximation
Let \(P_n\) be the set of polynomials of degree at most \(n\) and \(U_n=\{p\in P_n: x\in I \Rightarrow p(x)\in I\}\) where \(I=[0,1]\). A large part of the paper is devoted to finding a good parametrization for such bounded polynomials. This is first obtained by using the Lukács theorem for nonnegative polynomials and Euler's four squares identity which gives a \(2\pi\)-periodic solutions depending on three angles for \(n=1\) and by iteration for \(n>1\). This is used to prove that \(\inf_{p\in U_n}\|f-p\|\leq 2\inf_{p\in P_n}\|f-p\|\). Some ideas are given on how this can be generalized to bivariate polynomials. This requires to replace the four squares property by the eight squares identity of Degen. Some implementation details are given and illustrated with some numerical applications.
scientific article; zbMATH DE number 6812494
scientific article