The Lebesgue function for generalized Hermite-Fejér interpolation on the Chebyshev nodes

On Generalized Hermite–Fejér Interpolation of Lagrange Type on the Chebyshev Nodes

“…This is shown (using contour integration) in Byrne et al [3], and can also be demonstrated by equating coefficients of % &1 in the Laurent expansions about 0 of both sides of the identity…”

“…This result was obtained by Byrne et al [2], with a sharper version being developed in [3]. However, both results relied on the characterization of 4 2m, n as * 2m, n (\1), the proof of which is quite technical (see [2, pp. 351 357]).…”

An extension of the Grünwald–Marcinkiewicz interpolation theorem

“…There is no general theorem to bear out this expectation -indeed, in some specific cases, our expectations are not realised! (For example, see Byrne et al [1].) However, this theme has been the basis for many research investigations and so too it is in this paper.…”

“…Let Z + = {1,2,3,... }. Consider the infinite, triangular array of points in [-1,1] denoted by (1) T := (x k<n = cos((2fc -l)jr/(2n)) : * = 1, 2 , . .…”

“…In this section we prove Theorem 2 only for x in the open interval ( -1,1). Assume that the matrix of nodes is given by T as in(1) and that m e {1,3,5,... }. By (3), we define the Lebesgue constant associated with the operator H m<n by…”