CONVERGENCE OF DERIVATIVE OF ( 0 , 2 ) INTERPOLATORY POLYNOMIAL

In this paper, we consider the non-uniformly distributed zeros on the unit circle of nth Legendre polynomial. Here, we are interested to establish the convergence theorem for the derivative of (0,2) interpolatory polynomial on the above said nodes.


Introduction
J. Suranyi and P.Tur n [8] was first, who initiated the problem of Lacunary interpolation on zeros of , where, is the Legendre polynomial of degree .The problem of (0,2) interpolation on the roots of unity was first studied by O. Ki [7].He obtained its regularity, fundamental polynomials and established a convergence theorem for the same.Later on several Mathematicians have considered the Lacunary interpolation on the unit circle.After that author 1 (with K. K. Mathur) [1] considered the weighted (0,2) * -interpolation on the set of nodes obtained by projecting vertically the zeros of on the unit circle and established a convergence theorem for that interpolatory polynomial.Later on author 1 (with M. Shukla) [3] considered (0,2)-interpolation on the nodes, which are obtained by projecting vertically the zeros of on the unit circle, where stands for Jacobi polynomial, obtained the explicit forms and establish a convergence theorem for the same.Recently, authors [2] considered weighted (0,2)-interpolation on the nodes, which are obtained by projecting vertically the zeros of the onto the unit circle, established a convergence theorem for the same.
In 1990, K. Bal zs [5] proved the simultaneous convergence of the derivatives of Langrange interpolating polynomials by giving an estimate for by the aid of the Lebesgue constant of Lagrange interpolation.In 1993, K. Bal zs and T. Kilgore [6] considered the approximation of derivatives by interpolation.Also author 1 (with M. Shukla) [4], considered the convergence of the derivative of the Hermite interpolatory polynomial.
These have motivated us to consider the convergence of the derivative of the Lacunary interpolatory polynomial on the unit circle and established a convergence theorem.In section 2 we give some preliminaries and in section 3, we describe the problem.In section 4, we give the explicit formulae of the interpolatory polynomials.In section 5 and 6, estimation and convergence of interpolatory polynomials are given respectively.

Preliminaries
There are some well-known results in this section, which we shall use.
The differential equation satisfied by is We shall require the fundamental polynomials of Lagrange interpolation based on the zeros of and are respectively given as: (2.4) , (2.5) We will also use the following results For more details, one can see [9].

The Problem:
Let, satisfying: where, is a constant independent of and .

Convergence:
In this section, we prove the main theorem.

Conclusion:
Convergence of (0,2) Interpolation on non-uniformly distributed zeros on the unit circle by derivative of polynomial is strong than the convergence of the derivative of (0,2) Interpolation on non-uniformly distributed zeros on the unit circle by the same polynomial.