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 the weighted spaces L_p. If 1 ≤ p ≤ ∞, 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”.