SINGULARITY AND QUADRATURE REGULARITY OF (0, 1,…,m-2,m)-INTERPOLATION ON THE ZEROS OF (1-x) Pn-1(α,β) (x)

Abstract

In this paper, we show that if a problem of (0, 1,…,m-2,m)-interpolation on the zeros of (1-x)Pn-1α,β(x) (a > 1, β ≥ -1) has an infinity of solutions then the general form of the solutions is f0(x)+Cf(x) with an arbitrary constant C, where Pn-1(α,β) (x) stands for the (n-1)th Jacobi polynomial, and f0(x) and f(x) are fixed polynomials of degree ≤ mn-1, and, meanwhile, the explicit form of f(x) is given. Moreover, a necessary and sufficient condition of quadrature regularity of the interpolation in a manageable form is established.