On the non-existence of some interpolatory polynomials

Here we prove that ifxk,k=1,2,…,n+2are the zeros of(1−x2)Tn(x)whereTn(x)is the Tchebycheff polynomial of first kind of degreen,αj,βj,j=1,2,…,n+2andγj,j=1,2,…,n+1are any real numbers there does not exist a unique polynomialQ3n+3(x)of degree≤3n+3satisfying the conditions:Q3n+3(xj)=αj,Q3n+3(xj)=βj,j=1,2,…,n+2andQ‴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 wherePn(x)is the Legendre polynomial of degreen.

On A Problem of P. Turán on Lacunary Interpolation*

In 1955, Suranyi and P. Turán [8] considered the problem of existence and uniqueness of interpolatory polynomials of degree ≤ 2n-1 when their values and second derivatives are prescribed on n given nodes. Around this kind of interpolation - aptly termed (0, 2) interpolation - considerable literature has grown up since then. For more complete bibliography on this subject we refer to J. Balazs [3], Later we considered [10] the problem of modified (0, 2) interpolation when 2 the abscissas are the zeros of (1-x2) Tn(x), where Tn(x) is the Tchebycheff polynomial of the first kind (Tn(x) = cos n θ, x = cos θ).

Polynomials with some Prescribed Zeros

In connection with various problems concerning polynomialson the unit interval, the Tchebycheff polynomialis known to play a very important role [11, problem 34].