The aim of this book is to study certain weighted lacunary interpolation processes on an infinite interval. By lacunary interpolation we mean to study an interpolation process when non consecutive derivatives are prescribed on a given set of nodes. Here we have considered the existence, uniqueness, explicit representation of a modified weighted (0, 1, 3) and (0, 1, 2, ... , r - 2, r) interpolations on an arbitrary set of nodes, weighted (0, 2), (0, 1, 3) interpolations on the zeros of nth-Hermite polynomial and a mixed type (0; 0, 2) interpolation when function values and weighted (0, 2) are prescribed on the zeros of nth-Hermite polynomials and its derivative respectively. A convergence theorem has been obtained in these cases. The results obtained here are better to some earlier results obtained by several mathematicians in the sense that (i) an artificial looking condition, used for obtaining the explicit representation of the fundamental polynomials, has been replaced by a simple interpolatory condition (ii) the results have been obtained in the case when n is considered to be odd and (iii) an improved quantitative estimate of the interpolatory polynomials has been obtained.