Convergence of Hermite-Fejér interpolation over roots of third-kind Chebyshev polynomials

Article history: Received 29 April 2016 Received in revised form 10 July 2016 Accepted 10 August 2016 This paper considers the Hermite-Fejér interpolation to functions of bounded variation. This interpolation is considered when the nodes of interpolation are taken to be the roots of the third-kind Chebyshev polynomials. An estimate for the rate of convergence at the points of continuity for functions of bounded variations is given. It is also shown that, in this case, the rate of convergence cannot be improved asymptotically.


Introduction
*For a given function, the Lagrange interpolation is, in many cases, not satisfactory. It is known that there exists a continuous function whose Lagrange interpolation diverges everywhere. So, there is a need to introduce additional conditions either related to the properties of the function or considering interpolating also derivatives-related data; this kind of interpolation is called the Hermite interpolation and will be considered in section 3. If the function has, in particular, lack of derivative information, then we consider the derivatives equal to zeros; this kind of interpolation is called the Hermite-Fejér interpolation and will be considered in section 4. It is also known that when using the nodes of interpolation to be the roots of the orthogonal polynomials, then the convergence is faster and assured. The third-kind Chebyshev polynomials are introduced in section 2. For more on these topics, see (Szegö, 1959). The rate of convergence of the Hermite-Fejér interpolation on the roots of the third-kind Chebyshev polynomials is given in section 5. Conclusions are given in section 6.

Third-kind Chebyshev polynomials
The Chebyshev polynomials of the third kind, ( ), are defined to be the orthogonal polynomials over the interval [-1, 1] with respect to the weight function They satisfy the following orthogonality relations: They also satisfy the following recurrence relations: where 0 ( ) = 1, 1 ( ) = 2 − 1, ≥ 1.
They are also a special case of the Jacobi polynomials and are related to them by the formula: They also satisfy the following Rodrigues' type formula: They also satisfy the following second order differential equation: The roots of , = 1,2, … , of ( ) are all distinct real and lie in the interior of the interval. For more on the Chebyshev polynomials of the third kind, see (Doha et al., 2015, andAbd-Elhameed, 2014).
The Hermite polynomial 2 −1 ( ) is given explicitly by the formula: If the function ( ) is merely continuous on the interval [−1,1], Fejér investigated the case of setting ′ 2 −1 ( ) = ′ = 0, = 1,2, … , . In this case, the polynomial 2 −1 ( ) is called the step polynomial or the Hermite-Fejér interpolating polynomial which behaves more regularly than the Lagrange interpolating polynomials as becomes large. In the next section this polynomial is considered when the nodes of interpolation are the roots of the third-kind Chebyshev polynomials.

Hermite-Fejér interpolation
These roots are used with the Hermite-Fejér interpolation as nodes of interpolation. We require the polynomial to take on given values at these roots and also to fix the values of its derivative at these roots. Given a function ( ) on [−1,1], we want to find a polynomial of lowest possible degree that satisfies the following properties: where, see (Rababah, 2007), = cos( cos −1 2 ).

Rate of convergence
In this section, we consider the Hermite-Fejér interpolation for functions of bounded variation on the roots of the Chebyshev polynomials of third kind ( ). Let We estimate the rate of convergence of 2 −1 ( ) to ( ) at points of continuity of ( ). Let ∈ (−1,1), then we have and ( , ) has at most two elements, we get .
This can be rewritten as follows

The function ( ) is non-increasing and thus we have
This discussion leads to the following formula for the estimate of the rate of convergence in the following theorem. where [ , ] is the total variation of ( ) on [ , ].
To check the precision of the estimate, we consider the Hermite-Fejér interpolation for even to the function ( ) = 2 at = 0.

Conclusions
In this paper, we have considered the Hermite-Fejér interpolation to functions of bounded variation on the roots of the third-kind Chebyshev polynomials as the nodes of interpolation. An estimate for the rate of convergence at the points of continuity for functions of bounded variations is given in Theorem 1. It is also shown that the rate of convergence cannot be improved asymptotically in Theorem 2.