Modified (0, 2)-Interpolation on the Roots of Jacobi Polynomials. I (Explicit Formulae)

Abstract

Let the set of knots $$ - 1 = x_{n + 1} < x_n < ... < x_1 < x_0 = 1 (n \geqq 1)$$ (n ≧ 1) be given on the interval [-1, 1]. Find a polynomial Qm(x) of minimal degree satisfying (0, 2)-interpolational conditions at the inner knots and boundary conditions at the endpoints, that is $$Q_m^{(s)} (x_i ) = y_i^{(s)} (s = 0,2) for i = 1,..., u_1$$ and $$Q_m^{(j)} (x_O ) = \alpha _O^{(j)} for j = 1,..., k$$ $$Q_m^{(j)} (x_{n + 1} ) = \alpha _{n + 1}^{(j)} for j = 0,..., l$$ where yi(s),αO(j), αn+1(j) are arbitrarily given real numbers, and k, l are arbitrary fixed non-negative integers. In this paper the existence and uniqueness of the polynomial Qm(x) is proved if the inner nodal points are the zeros of Jacobi polynomials Pn2k + 1, 2l − 1 (x) or Pn2k − 1, 2l + 1 (x). Explicit formulae for the fundamental polynomials of interpolation are also given.