On the order of magnitude of fundamental polynomials of Hermite interpolation
The author deals with Hermite interpolation polynomials of the form \[ H_{mn} (f,x):= \sum_{k=1}^ n \sum_{j=0}^{m-1} f^{(j)} (x_{kn}) A_{jk}(x) \] for a function \(f\) that is \(m-1\) times continuously differentiable on the interval \([-1,1]\) (\(m\) an arbitrary positive integer) and a system of arbitrary interpolation nodes \(-1\leq x_{nn}< x_{n-1,n}< \cdots< x_{1n}\leq 1\). The fundamental polynomials \(A_{jk}(x)\) of degree at most \(mn-1\) satisfy the conditions \(A_{jk}^{(p)} (x_{qn})= \delta_{jp} \delta_{kq}\) for \(j,p= 0,\dots, m-1\) and \(k,q= 1,\dots, n\). As the main result of the paper, exact lower bounds for the quantities \(L_{jmn}:= \| \sum_{k=1}^ n | A_{jk}(x)| \|_ \infty\), \(j=0,\dots, m-1\), are established, namely \(L_{jmn}\geq c_ 1(\log n)/ n^ j\) if \(m-j\) is even, with \(c_ 1\) and \(c_ 2\) being positive constants depending only on \(j\) and \(m\). The Chebyshev nodes \(x_{kn}= \cos((2k-1) \pi/(2n))\) are used to show that these estimates are sharp.
scientific article; zbMATH DE number 508725