On the non-existence of some interpolatory polynomials

Abstract

Here we prove that if xk, k=1,2,…,n+2 are the zeros of (1−x2)Tn(x) where Tn(x) is the Tchebycheff polynomial of first kind of degree n, αj, βj, j=1,2,…,n+2 and γj, j=1,2,…,n+1 are any real numbers there does not exist a unique polynomial Q3n+3(x) of degree ≤3n+3 satisfying the conditions: Q3n+3(xj)=αj, Q3n+3(xj)=βj, j=1,2,…,n+2 and Q‴3n+3(xj)=γj, j=2,3,…,n+1. Similar result is also obtained by choosing the roots of (1−x2)Pn(x) as the nodes of interpolation where Pn(x) is the Legendre polynomial of degree n.