On Some Properties of Moduli of Smoothness with Jacobi Weights

We discuss some properties of the moduli of smoothness with Jacobi weights that we have recently introduced and that are defined as ωk,rφ(f(r),t)α,β,p:=sup0≤h≤t∥Wkhr∕2+α,r∕2+β(⋅)Δhφ(⋅)k(f(r),⋅)∥p,$$\displaystyle{\omega }_{k,r}^\varphi (f^{(r)},t)_{\alpha,\beta,p} :=\sup _{0\leq h\leq t}\left \|{\mathcal {W}}_{kh}^{r/2+\alpha,r/2+\beta }(\cdot )\Delta _{h\varphi (\cdot )}^k (f^{(r)},\cdot )\right \|{ }_{p},$$ where φ(x)=1−x2, Δhk(f,x) is the kth symmetric difference of f on [−1, 1], Wδξ,ζ(x):=(1−x−δφ(x)∕2)ξ(1+x−δφ(x)∕2)ζ,$$\displaystyle{\mathcal {W}}_\delta ^{\xi,\zeta } (x):= (1-x-\delta \varphi (x)/2)^\xi (1+x-\delta \varphi (x)/2)^\zeta,$$ and α, β > −1∕p if 0 < p < ∞, and α, β ≥ 0 if p = ∞. We show, among other things, that for all m, n∈ ℕ, 0 < p ≤∞, polynomials P_n of degree < n and sufficiently small t, ωm,0φ(Pn,t)α,β,p∼tωm−1,1φ(Pn′,t)α,β,p∼⋯∼tm−1ω1,m−1φ(Pn(m−1),t)α,β,p∼tm∥wα,βφmPn(m)∥p,$$\displaystyle\begin {array}{ll} {\omega }_{m,0}^{\varphi }(P_n, t)_{\alpha,\beta,p} & \sim t {\omega }_{m-1,1}^{\varphi }(P_n^{\prime }, t)_{\alpha,\beta,p} \sim \dots \sim t^{m-1}{\omega }_{1,m-1}^{\varphi }(P_n^{(m-1)}, t)_{\alpha,\beta,p} \\ & \sim t^m \left \|w_{\alpha,\beta } \varphi ^{m} P_n^{(m)}\right \|{ }_{p}, \end {array}$$ where w_α,β(x) = (1 − x)^α(1 + x)^β is the usual Jacobi weight. In the spirit of Yingkang Hu’s work, we apply this to characterize the behavior of the polynomials of best approximation of a function in a Jacobi weighted L_p space, 0 < p ≤∞. Finally we discuss sharp Marchaud and Jackson type inequalities in the case 1 < p < ∞.