International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 4, Pages 753-756
doi:10.1155/S016117128600090X
On the non-existence of some interpolatory polynomials
Department of Mathematics, California State University, Los Angeles 90032, California, USA
Received 20 May 1985
Copyright © 1986 C. H. Anderson and J. Prasad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.