Weighted D-T moduli revisited and applied

Abstract

$ f\in L_p[-1,1]\cap C^{r-1}(-1,1)$ , $ r\ge1$ , that have an $ (r-1)$ st absolutely continuous derivative in $ (-1,1)$ and such that $ \varphi^rf^{(r)}$ is in $ L_p[-1,1]$ , where $ \varphi(x)=(1-x^2)^{1/2}$ . These moduli are equivalent to certain weighted D-T moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted moduli to weighted approximation, which was the purpose of the original D-T moduli, we apply these moduli to obtain Jackson-type estimates on the approximation of functions in $ L_p[-1,1]$ (no weight), by means of algebraic polynomials. We also have some inverse theorems that yield characterization of the behavior of the derivatives of the function by means of its degree of approximation.