A Study of Certain Interpolatory Processes on Unit Circle : Interpolation on Unit Circle

Hermite Interpolation is a particular case in which we are given only the values of function and its first derivative at the given set of points in a given interval. As against this, we shall use the term "Lacunary Interpolation", in which the value of the function and its derivatives of higher orders are prescribed at the given set of points. In this, we have considered the problem of existence, explicit representation, estimation and convergence behavior of several interpolatory polynomials on the nodes, which are obtained by projecting vertically the zeros of nth Legendre polynomial together with +1 and -1 on unit circle. Weighted Lacunary interpolation, say weighted (0, 2) and (0, 1, 3) interpolations are also considered on unit circle. Various other types of interpolation have also been investigated. Finally, we obtained the rate of convergence and convergence theorems of such interpolatory polynomials.