PAL-TYPE HERMITE INTERPOLATION ON INFINITE INTERVAL

For given arbitrary numbers alpha(k,n), 1 less than or equal to k less than or equal to n, and beta(k,n), 1 less than or equal to k less tha n or equal to n - 1, we seek to determine explicitly polynomials R(n)( x) of degree at most 2n - 1 (n even), given by [GRAPHICS] such that R( n)(x(k,n)) = alpha(k,n), k = 1(1)n R'(n)(y(k,n)) = beta(k,n), k = 1(1) n-1 and [GRAPHICS] where l(k,n)(x) are fundamental functions of Lagran ge interpolation, {x(k,n)}(n - 1)(k = 1) are the zeros of H-n(1)(x). L et the interpolated function f be continuously differentiable, satisfy ing the conditions. Lim x(2r)f(x)p(x) = 0, r = 0, 1, 2,..., \x\-->infi nity and Lim p(x)f'(x) = 0, where p (x) = e(-x2/2). \x\-->infinity Fur ther, taking alpha(k,n)f(x(k,n)) k = 1(1)n, and beta(k,n) = f(y(k,n)), k = 1(1)n-1, in the first equation, then for the sequence of interpol atory polynomials R(n)(n = 2, 4,...) we have the estimate [GRAPHICS] w hich holds on the whole real line; O does not depent on n and x and om ega is the modulus of continuity of f' introduced by Greud. (C) 1995 A cademic Press, Inc.