Estimates of divided differences of real-valued functions defined with a noise (Q2912217)

For \(k\in \mathbb N\), let \(X_k\) be the set of all real vectors \(x=(x_j)_{j=0}^k\) with \(x_j < x_{j+1}\). Let \(Y_k\) be the set of all real vectors \(y = (y_j)_{j=0}^k\) with \(|y_j|\leq 1\). Further, \(F(x,y)\) denotes the set of all real functions \(f\) with \(f(x_j) = y_j\) \((j=0,\dots,k)\). For \(z =(z_j)_{j=0}^k \in X_k\), the \(k\)-th order divided difference of \(f\) is \([z_0,\dots,z_k]f\). For given \(x\in X_k\) and \(y\in Y_k\), denote NEWLINE\[NEWLINE \lambda_k(x,y) := \inf_{f\in F(x,y)}\, \sup_{z\in X_k} |[z_0,\dots,z_k]f|\,. NEWLINE\]NEWLINE In this paper, the authors estimate the values NEWLINE\[NEWLINE \lambda_k (x) := \sup_{y\in Y_k} \lambda_k (x,y)\,,\quad \mu_k(x) := {\mathbb E}(\lambda_k (x,y))\,. NEWLINE\]NEWLINE Here \({\mathbb E}(\lambda_k (x,y))\) is the expectation of \(\lambda_k (x,y)\) taken over \(y=(y_j)_{j=0}^k\in Y_k\), where the entries \(y_j\) are independent and uniformly distributed on \([-1,\,1]\). In the case \(x_j\in [-1,\,1]\), the Chebyshev nodes are optimal for the worst-case error \(\lambda_k(x)\), but they are not optimal for the mean error \(\mu_k(x)\).