Convergence of Cubic Spline Super Fractal Interpolation Functions

Abstract: In the present work, the notion of Cubic Spline Super Fractal Interpolation Function (SFIF) is introduced to simulate an object that depicts one structure embedded into another and its approximation properties are investigated. It is shown that, for an equidistant partition points of [x_0,x_N], the interpolating Cubic Spline (SFIF) and their derivatives converge respectively to the data generating function and its derivatives at the rate of h^2-j+e (0<e<1), j=0,1,2 as the norm h of the partition of [x_0,x_N] approaches zero. The convergence results for Cubic Spline (SFIF) found here show that any desired accuracy can be achieved in the approximation of a regular data generating function and its derivatives by a Cubic Spline (SFIF) and its corresponding derivatives.