On moduli of smoothness with Jacobi weights

The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights $(1-x{)}^{\alpha }(1+x{)}^{\beta }$ for functions in the Jacobi weighted $L_p[-1,1]$, $0<p\le \mathrm{\infty }$, spaces. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted $L_p$ spaces. If $1\le p\le \mathrm{\infty }$, then these moduli are equivalent to certain weighted $K$-functionals (and so they are equivalent to certain weighted Ditzian-Totik moduli of smoothness for these $p$), while for $0<p<1$ they are equivalent to certain “Realization functionals”.