The paper is devoted to studying a Pál-type interpolation problem on the roots of Ultraspherical polynomials of degree n-1 with parameter k+1 on the closed interval -1 to 1. The aim of this paper is to find a unique interpolatory polynomial of degree at most m equal to 2n+2k+1 satisfying the interpolatory conditions that is, function values of the polynomial of degree m at the zeros of the function values of the ultraspherical polynomials and the first derivative values of the polynomial of degree m at the zeros of the first derivative values of the ultraspherical polynomials.We will use the special type of Hermite-boundary conditions at the end points of interval -1 to 1, which are defined by, the lth derivative of the polynomial of degree m at the zeros of the boundary point 1, where l goes from 0 to k+1 and the lth derivative of the polynomial of degree m at the zeros of the boundary point -1, where l goes from 0 to k+2. Further, we will prove the existence,uniqueness and explicit representation of the interpolatory polynomial. For, the prove of order of convergence of the interpolatory polynomial, we will prove the order of convergence of the first derivative of the first kind fundamental polynomials and order of convergence of the first derivative of the second kind fundamental polynomials.
Srivastava, R. and Singh, Yamini
An Interpolation Process on the Roots of Ultraspherical Polynomials,
Applications and Applied Mathematics: An International Journal (AAM), Vol. 13,
2, Article 32.
Available at: https://digitalcommons.pvamu.edu/aam/vol13/iss2/32