No abstract

In this paper we have discussed a special lacunary interpolation problem in which the function values, first derivatives at the nodes and the third derivatives at any point λ (0 in between the nodes are prescribed. We have solved the unique existence and convergence problems, using spline functions. As this holds for any λ (0 we named it a generalized problem.

In the present paper a special lacunary interpolation problem is solved in which function value, first derivatives and fourth derivatives are prescribed at nodes of the unit interval I = [0, 1]. A special spline function is obtained for it. Then the theorem of unique existence and convergence for this spline function are proved. In our next communication we will show that this special function can be used to solve Cauchy's Initial value problem.

Let f be a real valued function defined in [0, 1], with values known at intermediate points such that the first derivatives of f at all nodes are also known at intermediate points.In this paper, we construct an interpolatory quartic spline s which interpolates the function f. Unique existence and convergence of this spline are also established. This type of construction is known to have found aesthetic utility in finding areas under or bounded by polynomial curves.