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A note on Hermite-Fejer Interpolation on the non-uniform nodes of the unit circle.
Abstract
This paper deals with the Hermite-Fejér interpolation problem on the unit circle with the nodal system containing the vertically projected zeros of Jacobi's polynomial with boundary points on the unit circle. We worked upon three nodal structures throughout this paper and obtained rate of convergence for each case. Moreover, we did a comparison of all the three cases and provided some important conclusions.
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The aim of this paper is to study a Lagrange-Hermite interpolation on the nodes, which are obtained by projecting vertically the zeroes of the (1-x^2 )P_n (x) on the unit circle, where P_n (x) stands for n^th Legendre polynomial. We prove the regularity of the problem, give explicit forms and establish a convergence theorem for the same.
The Chebyshev nodal systems play an important role in the theory of Hermite interpolation on the interval [-1,1][-1,1]. For the cases of nodal points corresponding to the Chebyshev polynomials of the second kind Un(x)Un(x), the third kind Vn(x)Vn(x) and the fourth kind Wn(x)Wn(x), it is usual to consider the extended systems, that is, to add the endpoints −1 and 1 to the nodal system related to Un(x)Un(x), to add −1 to the nodal system related to Vn(x)Vn(x) and to add 1 to the nodal system related to Wn(x)Wn(x). The interpolation methods that are usually used in connection with these extended nodal systems are quasi-Hermite interpolation and extended Hermite interpolation, and it is well known that the performance of these two great methods is quite good when it comes to continuous functions.
This work attempts to complete the theory concerning these extended Chebyshev nodal systems. For this, we have obtained a new formulation for the Hermite interpolation polynomials based upon barycentric formulas. The feature of this approach is that the derivatives of the function at the endpoints of the interval are also prescribed. Further, some convergence results are obtained for these extended interpolants when apply to continuous functions.
The paper deals with the order of convergence of the Laurent polynomials of Hermite-Fejér interpolation on the unit circle with nodal system, the roots of a complex number with modulus one. The supremum norm of the error of interpolation is obtained for analytic functions as well as the corresponding asymptotic constants.
In order to approximate functions defined on the real semiaxis, we introduce a new operator of Hermite–Fejér-type based on Laguerre zeros and prove its convergence in weighted uniform metric.
This paper is devoted to studying an interpolation problem on the circle, which can be considered an intermediate problem between Lagrange and Hermite interpolation. The difference as well as the novelty is that we prescribe Lagrange values at the roots of a complex number with modulus one and we prescribe values for the first derivative only on half of the nodes. We obtain two types of expressions for the interpolation polynomials: the barycentric expressions and another one given in terms of an orthogonal basis of the corresponding subspace of Laurent polynomials. These expressions are very suitable for numerical computation. Moreover, we give sufficient conditions in order to obtain convergence in case of continuous functions and we obtain the rate of convergence for smooth functions. Finally we present some numerical experiments to highlight the results obtained.
In this paper we deal with Hermite interpolation problems on the unit circle considering up to the second derivative for the interpolation conditions and taking equally spaced points as nodal system. In the extended Fejér case, which corresponds to take vanishing values for the first two derivatives, we prove the uniform convergence for the interpolants related to continuous functions with smooth modulus of continuity. We also consider the Hermite case with non vanishing conditions for the derivatives for which we establish sufficient conditions on the interpolation conditions to obtain convergence.
The Hermite-Fejér interpolation polynomial H n [ f ] of degree ≤2 n —1 is defined by
⁽¹⁾
Where
⁽²⁾
are the zeroes of Chebyshev polynomial of first kind T n (x) =cos n (arc cos x ).
In this paper, we study the convergence of the Hermite–Fejér and the Hermite interpolation polynomials, which are constructed by taking equally spaced nodes on the unit circle. The results that we obtain are concerned with the behaviour outside and inside the unit circle, when we consider analytic functions on a suitable domain. As a consequence, we achieve some improvements on Hermite interpolation problems on the real line. Since the Hermite–Fejér and the Hermite interpolation problems on [−1,1], with nodal systems mainly based on sets of zeros of orthogonal polynomials, have been widely studied, in our contribution we develop a theory for three special nodal systems. They are constituted by the zeros of the Tchebychef polynomial of the second kind joint with the extremal points −1 and 1, the zeros of the Tchebychef polynomial of the fourth kind joint with the point −1, and the zeros of the Tchebychef polynomial of the third kind joint with the point 1. We present a simple and efficient method to compute these interpolation polynomials and we study the convergence properties.
In this note, an extension to the unit circle of the classical Hermite-Fejér Theorem is given.
- S Bahadur
- Varun
Bahadur, S. and Varun, Convergence of Interpolatory polynomial between Lagrange and Hermite, Annals of Pure and Applied Mathematics, Vol., 17(1)
(2018).
On Theory of Interpolation
- D L Berman
Berman, D.L., On Theory of Interpolation, Soviet Math.Dokl, 6 (1965), 945-948.