The Lebesgue Constant for Higher Order Hermite-Fejér Interpolation on the Chebyshev Nodes Original Research Article

For a fixed integer m ≥ 0, and for n = 1, 2, 3, ..., let λ2m, n(x) denote the Lebesgue function associated with (0, 1,..., 2m) Hermite-Fejér polynomial interpolation at the Chebyshev nodes {cos[(2k−1) π/(2n)]: k=1, 2, ..., n}. We examine the Lebesgue constant Λ2m, n ≔ max{λ2m, n(x): −1 ≤ x ≤ 1}, and show that Λ2m, n = λm, n(1), thereby generalising a result of H. Ehlich and K. Zeller for Lagrange interpolation on the Chebyshev nodes. As well, the infinite term in the asymptotic expansion of Λ2m, n) as n → ∞ is obtained, and this result is extended to give a complete asymptotic expansion for Λ2, n.