Approximation by rational functions with prescribed numerator degree in Lp spaces for 1 p∞

The present paper establishes a complete result on approximation by rational functions with prescribed numerator degree in LP spaces for 1 < p < infinity and proves that if f (x) is an element of L-[-1,1](p) changes sign exactly l times in (-1,1), then there exists r(x) is an element of R-n(1) such that parallel tof(x) - r(x)parallel to(Lp) less than or equal to C(p,l,b)omega(f,n(-1))(Lp), where R-n(l) indicates all rational functions whose denominators consist of polynomials of degree n and numerators polynomials of degree l, and C-p,C-l,C-b is a positive constant depending only on p, l and b which relates to the distance among the sign change points of f(x) and will be given in 3.