On the preservation of global smoothness by some interpolation operators (Q2714352)

Let \(\text{Lip}_{M}\left(\alpha;[a,b]\right):=\left\{ f:[a,b]\rightarrow \mathbb{R}:|f(x) -f(y)|\leq M|x-y|^{\alpha},\forall x,y\in[a,b] \right\}\), for \(0<\alpha\leq 1\). If \(f\in \text{Lip}_{M}\left(\alpha;[a,b] \right)\) and \(\left\{L_{n}(f)\right\}_{n\geq 1}\) is a sequence of approximation operators associated to \(f\), the partial preservation of the smoothness consists in checking the validity of an implication of the form: NEWLINE\[NEWLINE f\in \text{Lip}_{M}\left(\alpha;[a,b]\right)\Rightarrow L_{n}(f) \in \text{Lip}_{M'}\left(\alpha;[a,b]\right),n\in\mathbb N,\tag{1}NEWLINE\]NEWLINE or of the stronger condition: NEWLINE\[NEWLINE \omega\left(L_{n}(f);h\right)\leq c \omega(f;h), 0\leq h\leq h_{0}, n\in\mathbb{N}.\tag{2}NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEThe authors obtain positive results as well as negative ones, for the case when the approximation operators are of interpolatory type. For instance, the sequence of Hermite-Fejér interpolation polynomials for the knots \(x_{k+1}:=-1+2(k-1)/(n-1)\), \(k=1,2,\dots,n\), from \([-1,1]\), has the property that \(f\in \text{Lip}_{1}(1;[-1,1])\) implies that there is neither \(\alpha\in(0,1)\) nor \(M>0\) such that \(H_{2n-1}(f)\in \text{Lip}_{M} \left(\alpha;[-1,1]\right)\), \(n\in\mathbb{N}\). NEWLINENEWLINENEWLINEThere are given positive results for the implication (1) and the inequality (2) for the Hermite-Fejér polynomials on the Chebyshev knots of the first kind, for the Lagrange interpolation polynomials on the Chebyshev knots of the second kind and \(\pm 1\), and for the Shepard interpolation polynomials.