Oscillation-diminishing polynomial approximation in Cw0 and Lp(R,dμ)Lp(R,dμ), 1≤p<∞1≤p<∞

Let 1≤p<∞1≤p<∞. We show that ‘positive polynomial approximation property’ holds in the space Lp(R,dμ)Lp(R,dμ) (or Cw0) if and only if the algebraic polynomials are dense in L2p(R,dμ)L2p(R,dμ) (or Cw0). If μμ is not a 2p2p-singular measure (or w is not a singular weight), this also implies the more general ‘oscillation-diminishing polynomial approximation property’.