The Lebesgue constant for higher order Hermite-Fejér interpolation on the Chebyshev nodes
Let \(\lambda_{2m, n} (x)\) denote the Lebesgue function associated with \((0, 1,\dots, 2m)\) Hermite-Fejér polynomial interpolation on the Chebyshev nodes \(x_{k,n}:= \text{cos} [(2k-1) \pi/2n]\), \(k=1, \dots, n\). Here \(m\geq 0\) and \(n\) runs over positive integers. Then \(\Lambda_{2m, n}:= \max_{|x|\leq 1} \lambda_{2m, n} (x)\) is the Lebesgue constant. The authors show that \(\Lambda_{2m, n}= \lambda_{2m, n} (1)\), thereby generalizing a result of \textit{H. Ehlich} and \textit{K. Zeller} [Math. Ann. 164, 105-112 (1966; Zbl 0136.046)], who proved the case when \(m=0\), the case of Lagrange interpolation on Chebyshev nodes. The study of Lebesgue constant for Lagrange and Hermite interpolation has a respectable history. Here the authors also obtain the asymptotic result (Theorem 2) that as \(n\to \infty\), \[ \Lambda_{2m, n}= {{2(2m)!} \over {\pi 2^{2m} (m!)^2}} \log n+ O(1) \] which makes precise an earlier result of \textit{R. Sakai} and \textit{P. Vertesi} [Stud. Sci. Math. Hung. 28, 87-97 (1993; Zbl 0802.41006); ibid. 28, 379-386 (1993; reviewed below)]. They also give an asymptotic expansion for \(\Lambda_{2,n}\) on Chebyshev nodes, which is analogous to a similar result by \textit{P. N. Shivakumar} and \textit{R. Wong} [Math. Comput. 39, 195-200 (1982; Zbl 0492.41048)]\ for \(\Lambda_{0, n}\), the case of Lagrange interpolation on Chebyshev nodes. For fuller details the reader should refer to the paper itself.
scientific article; zbMATH DE number 803681