A constructive approach to cubic Hermite Fractal Interpolation Function and its constrained aspects

• Published in: BIT Numerical Mathematics Vol. 53; no. 4; pp. 841 - 865
• Main Authors: Chand, A. K. B., Viswanathan, P.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.12.2013
• Subjects: Computational Mathematics and Numerical Analysis; Mathematics; Mathematics and Statistics; Numeric Computing; Fractal Interpolation Function; Positivity; 28A80; 65D05; Iterated Function System; 65D17; Shape preservation; 65D07; Cubic spline; 41A29
• ISSN: 0006-3835; 1572-9125
• DOI: 10.1007/s10543-013-0442-4

The theory of splines is a well studied topic, but the kinship of splines with fractals is novel. We introduce a simple explicit construction for a -cubic Hermite Fractal Interpolation Function (FIF). Under some suitable hypotheses on the original function, we establish a priori estimates (with respect to the L p -norm, 1≤ p ≤∞) for the interpolation error of the -cubic Hermite FIF and its first derivative. Treating the first derivatives at the knots as free parameters, we derive suitable values for these parameters so that the resulting cubic FIF enjoys global smoothness. Consequently, our method offers an alternative to the standard moment construction of -cubic spline FIFs. Furthermore, we identify appropriate values for the scaling factors in each subinterval and the derivatives at the knots so that the graph of the resulting -cubic FIF lies within a prescribed rectangle. These parameters include, in particular, conditions for the positivity of the cubic FIF. Thus, in the current article, we initiate the study of the shape preserving aspects of fractal interpolation polynomials. We also provide numerical examples to corroborate our results.