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,\dots,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 $f^{(k)}(x_0)$ and $f^{(k)}(x_N)$, $k=1,2,\dots,r,$ are prescribed, our method of construction of the $C^r$-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 $f_{\Delta}(x)$ through moments is developed. It is shown that the sequence $\{f_{\Delta_k}(x)\}$ converges to the defining data function $\Phi(x)$ on two classes of sequences of meshes at least as rapidly as the square of the mesh norm $\| \Delta_k \|$ approaches to zero, provided that $\Phi^{(r)}(x)$ is continuous on I for r=2,3, or 4.