Generalized Cubic Spline Fractal Interpolation Functions

We construct a generalized Cr-Fractal Interpolation Function (Cr-FIF) f by prescribing any combination of r values of the derivatives f (k), k = 1, 2, . . . , r, at boundary points of the interval I = [x0, xN ]. Our approach to construction settles several questions of Barnsley and Harrington [J. Approx Theory, 57 (1989), pp. 14–34] when construction is not restricted to prescribing the values of f (k) at only the initial endpoint of the interval I. In general, even in the case when r equations involving f (x0) and f (xN ), k = 1, 2, . . . , r, are prescribed, our method of construction of the Cr-FIF works equally well. In view of wide ranging applications of the classical cubic splines in several mathematical and engineering problems, the explicit construction of cubic spline FIF fΔ(x) through moments is developed. It is shown that the sequence {fΔk (x)} converges to the defining data function Φ(x) on two classes of sequences of meshes at least as rapidly as the square of the mesh norm ‖Δk‖ approaches to zero, provided that Φ(r)(x) is continuous on I for r = 2, 3, or 4.