Discretization for uniform polynomial approximation
Let P be the polynomial of degree less than or equal to n which is the best approximation to a given f in C[-1,1]. An approximation to P can be computed by choosing a finite subset F of [-1,1] and calculating the polynomial \(P_ F\), of degree less than or equal to n, which best approximates f on F. Then if \(| F|\) is small, estimates show that the discretization error, as measured by \(\| P-P_ F\|\), is also small. A classical choice for the set F of m points is (1) \(\{\) cos((2j- 1)\(\pi\) /2m): \(j=1,2,...,m\}\). A natural formulation of this discretization problem is developed which leads to a specific criterion for the choice of points in F. It is shown that, by this criterion, the choice of points in (1) is asymptotically best, but not best.
scientific article; zbMATH DE number 3844302
scientific article