On moduli of smoothness with Jacobi weights - Kopotun, K. A.; Leviatan, D.; Shevchuk, I. A. (Ukrains’kyi Matematychnyi Zhurnal, 2018)
We introduce the moduli of smoothness with Jacobi weights $(1 x)α (1+x)β$ for functions in the Jacobi weighted spaces$L_p[ 1, 1],\; 0 < p ≤ ∞ $. These moduli are used to characterize the smoothness of (the derivatives of) functions in theweighted spaces $L_p$. If $1 ≤ p ≤ ∞$, then these moduli are equivalent to certain weighted $K$-functionals (and so they areequivalent to certain weighted Ditzian – Totik moduli of smoothness for these $p$), while for $0 < p < 1$ they are equivalentto certain “Realization functionals”.