Sign-Preserving Approximation of Periodic Functions

Abstract

We prove the Jackson theorem for a zero-preserving approximation of periodic functions (i.e., in the case where the approximating polynomial has the same zeros y _i) and for a sign-preserving approximation [i.e., in the case where the approximating polynomial is of the same sign as a function f on each interval (y _i, y _{i − 1})]. Here, y _i are the points obtained from the initial points −π ≤ y _2s ≤_{y 2s}−1 <...< y₁ < π using the equality yi = y_{i + 2s} + 2π furthermore, these points are zeros of a 2π-periodic continuous function f.