Some interpolatory properties of Hermite polynomials

Abstract

1. Professor TURAN and his colleagues in a series of papers on interpolation have discussed i) the problem of existence and uniqueness, it) the problem of explicit representation and iii) the problem of convergence for (0, 2)-interpolation by taking as abscissae the zeros of H , ( x ) = (l--x2)P;;_l(x) where P~-l(x) is the Legendre polynomial of degree _<--n--1. By (0, 2)-interpolation we mean the construction of a polynomial of degree ~ 2 n l , when the value of the function and its second derivative at the zeros of Hn(x) are prescribed. Later on SaXENA and SHARMA [3] have studied the aforesaid problems for (0, 1, 3)-interpolation taking the same abscissae as those used by P. TURAN. Later SAXENA [4] has extended the results to (0, 1, 2, 4)-interpolation. The object of this note is to consider the problem of existence and uniqueness and of explicit representation for (0, 2)and (0, 1, 3)-interpolation, respectively, choosing the abscissae as the zeros of H,~(x), the Hermite polynomial of degree n, which are given by