Pál-Type Hermite Interpolation on Infinite Interval

For given arbitrary numbers αk,n, 1 ≤ k ≤ n, and βk,n, 1 ≤ k ≤n − 1, we seek to determine explicitly polynomials Rn(x) of degree at most 2n − 1 (n even), given by [formula] such that Rn(xk,n) = αk,n, k = 1(1)nR′n(yk,n) = βk,n, k = 1(1)n − 1 and [formula] where lk,n(x) are fundamental functions of Lagrange interpolation, {xk,n}nk = 1 are the zeros of the nth Hermite polynomial Hn(x), and {yk,n}n − 1k = 1 are the zeros of H1n(x). Let the interpolated function ƒ be continuously differentiable, satisfying the conditions: [formula] and [formula] Further, taking αk,n = ƒ(xk,n) k = 1(1)n, and βk,n = ƒ(yk,n), k = 1(1)n − 1, in the first equation, then for the sequence of interpolatory polynomials Rn(n = 2, 4,…) we have the estimate [formula] which holds on the whole real line; O does not depent on n and x and ω is the modulus of continuity of ƒ′ introduced by Freud.