“…In the case of Hermite interpolation, a theorem on extreme nodes was established by the authors that agreed with Pottinger's results [32] and was an improvement of the results obtained in [25]. Pottinger [33] used the Chebyshev nodes to prove that the convergence condition depends on the norms of H 2n+1 . The author also proved that the growth of the operator norms is of n th order:…”
Section: Literature Review On Hermite Polynomialssupporting
With progress on both the theoretical and the computational fronts, the use of Hermite interpolation for mathematical modeling has become an established tool in applied science. This article aims to provide an overview of the most widely used Hermite interpolating polynomials and their implementation in various algorithms to solve different types of differential equations, which have important applications in different areas of science and engineering. The Hermite interpolating polynomials, their generalization, properties, and applications are provided in this article.
“…In the case of Hermite interpolation, a theorem on extreme nodes was established by the authors that agreed with Pottinger's results [32] and was an improvement of the results obtained in [25]. Pottinger [33] used the Chebyshev nodes to prove that the convergence condition depends on the norms of H 2n+1 . The author also proved that the growth of the operator norms is of n th order:…”
Section: Literature Review On Hermite Polynomialssupporting
With progress on both the theoretical and the computational fronts, the use of Hermite interpolation for mathematical modeling has become an established tool in applied science. This article aims to provide an overview of the most widely used Hermite interpolating polynomials and their implementation in various algorithms to solve different types of differential equations, which have important applications in different areas of science and engineering. The Hermite interpolating polynomials, their generalization, properties, and applications are provided in this article.
“…Using Theorems 1 and 2, we havehere . By (1.1) and the formula (see, Pottinger 1978, p. 272),where is the best approximation to , we haveIn view of , using Jackson’s theorem (Natanson 1965, p. 86), we havewhich completes the proof of Theorem 3.
…”
Section: The Main Convergence Resultsmentioning
confidence: 57%
“…Before we state our theorem, we define where is the space of all polynomials of degree . (this definition is stated in Pottinger 1978, p. 272).…”
In this paper, we investigate the simultaneous approximation of a function f(x) and its derivative by Hermite interpolation operator based on Chevyshev polynomials. 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).
“…( cos(2nj)0, sin(2n-j)0, Similarly, using the identity sin nora sin nOk + sin join sin jOk 3=1 k,m 1, rt, (3 6) we can prove (3 4) satisfies (3 ,5S,(.f, COSOk) ,5f(cosOk) (k 1,...,n)…”
Section: Fundamental Polynomials Of the Interpolationmentioning
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