Pointwise and global estimate for approximation by rational functions with prescribed numerator degree

For any nonnegative continuous function f(x)f(x) defined on [−1,1][−1,1], and f≢0f≢0, the present paper proves that, there is a polynomial Pn(x)∈ΠnPn(x)∈Πn such that |f(x)−1Pn(x)|≤C(λ)ωφλ(f,n−1δn1−λ(x)), where δn(x)=1−x2+1/n,0≤λ≤1, and ΠnΠn is the set of all polynomials of degree nn. When f(x)f(x) has finite many sign change points, say ll points, we also construct a rational function r(x)∈Rnl such that |f(x)−r(x)|≤C(λ,δ,l)ωφλ(f,n−1δn1−λ(x)).