Zur Hermite‐Interpolation

Convergence and norm estimates of Hermite interpolation at zeros of Chevyshev polynomials

“…Since and , using the fact that , the result follows, which agrees with Pottinger’s estimate (Pottinger 1976). …”

“…We shall prove that when the nodal matrix M consists of the mixed Chevyshev nodes, the formula (2) also holds for . In the last section, we establish a general theorem on extreme points for Hermite operator that agrees with a known result by Pottinger (1976). …”

“…We also establish general theorem on extreme points for Hermite interpolation operator. Some results are considered to be an improvement over those obtained in Al-Khaled and Khalil (Numer Funct Anal Optim 21(5–6): 579–588, 2000), while others agrees with Pottinger’s results (Pottinger in Z Agnew Math Mech 56: T310–T311, 1976). …”

“…(21). Al-Khaled and Khalil (2000), also Pottinger (1976), they studied the Hermite type interpolation and gave some estimates for norms of certain interpolation operators on the space of continuously differentiable functions on I . In this section, we study the Hermite interpolation operator given by Eq.…”

“…Given a positive integer n , and a function f , we denote by the error of best uniform approximation of f by algebraic polynomials of degree at most n . Pottinger (1976) proved that when the nodal matrix M consists of the Chevyshev nodes, the convergenceholds for . There is an extensive literature on Hermite interpolation (see, for example Al-Khaled 2000; Agarwal and Wong 1991; Sababhah et al.…”