On moduli of smoothness with Jacobi weights

Abstract

We introduce the moduli of smoothness with Jacobi weights $(1 x)\alpha (1+x)\beta$ for functions in the Jacobi weighted spaces $L_p[ 1, 1],\; 0 < p \leq \infty $. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted spaces $L_p$. If $1 \leq p \leq \infty$, 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”.