On the non-existence of some interpolatory polynomials

C. H. Anderson and J. Prasad

International Journal of Mathematics and Mathematical Sciences, 1986, vol. 9, 1-4

Abstract:

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

Date: 1986
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:327450

DOI: 10.1155/S016117128600090X

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