RADAU QUADRATURE FOR AN ALMOST QUASI-HERMITE-FEJÉR-TYPE INTERPOLATION IN RATIONAL SPACES

In this paper, we have studied an almost quasi Hermite-Fejér-type interpolation in rational spaces. A Radau type quadrature formula has also been obtained for the same.


Introduction
Hermite Fejér and Quasi-Hermite-Fejér-type interpolation processes has been a subject of interest for several mathematicians. In almost all the cases the interpolatory polynomials are considered on the nodes which are the zeros of certain classical orthogonal polynomials. The main idea of the present paper is to construct a rational interpolation process and its corresponding quadrature formula with prescribed nodes based on the Chebyshev Markov fractions.
In 1962, Rusak [9] initiated the study of interpolation processes by means of rational functions on the interval [−1, 1]. The nodes were taken to be the zeros of Chebyshev-Markov rational fractions. In [6] rational interpolation functions of Hermite-Fejér-type were constructed [7]. Min [4] was the first to consider the rational quasi-Hermite-type interpolation. He constructed the interpolatory function and proved its uniform convergence for the continuous functions on the segment with the restriction that the poles of the approximating rational functions should not have limit points on the interval [−1, 1].
Recently, based on the ideas of [6] and using method that was different from that of [4], Rouba et. al. ( [5], [8]) revisited the rational interpolation functions of Hermite-Fejér-type. They also proved the uniform convergence of the interpolation process for the function f ∈ C[−1, 1] and obtained explicitly its corresponding Lobatto type quadrature formula.
In this paper, we have considered an almost quasi-Hermite-Fejér-type interpolation process on the zeros of the rational Chebyshev-Markov sine fraction on the semi closed interval (−1, 1], that is, when the interpolatory condition is prescribed only at one of the end points. A Radau type quadrature formula corresponding to the interpolation process has also been obtained.

Preliminaries
Consider a set of points a k , k = 0, 1, · · · , 2n − 1 which are either real and a k ∈ (−1, 1) or be paired by complex conjugation. Also let U n (x) be the rational Chebyshev-Markov sine fraction, .
The rational fraction U n (x) can be expressed as where P n−1 (x) is an algebraic polynomial of degree n − 1 with real coefficient. The fraction U n (x) has n − 1 zeros on the interval (−1, 1) given by, where µ 2n (x) is given by (2.2). Also, the rational function λ 2n (x), given by (2.4), can be expressed as where q 2n−1 (x) is a polynomial of degree atmost 2n − 1. It has no zeros on [−1, 1].

Radau-type quadrature formula
For a given function f defined on [−1, 1], we define the function .

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.