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International Journal of Mathematics and Mathematical Sciences
Volume 9, Issue 4, Pages 753-756

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 Hindawi Publishing Corporation. 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.


Here we prove that if xk, k=1,2,,n+2 are the zeros of (1x2)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 Q3n+3(xj)=γj, j=2,3,,n+1. Similar result is also obtained by choosing the roots of (1x2)Pn(x) as the nodes of interpolation where Pn(x) is the Legendre polynomial of degree n.