A New Class of Rational Cubic Fractal Splines for Univariate Interpolation

Abstract

Fractal interpolation functions that share smoothness or nonsmoothness property of the prescribed interpolation data provide a novel method of interpolation. The present paper proposes a new type of rational cubic spline fractal interpolation function which involves two families of free shape parameters and which does not require derivatives at knots for its construction. The scaling factors inherent with the structure facilitate the proposed rational fractal interpolation function to recapture a classical rational cubic spline studied earlier in the literature as a special case. In addition, the scaling factors are the key ingredients that provide fractality to the derivative of the constructed interpolant. Thus, in contrast to the classical nonrecursive rational splines, the proposed rational cubic fractal spline can produce interpolants whose derivatives have irregularity in finite or dense subsets of the interpolation intervals depending on the nature of the problem. Assuming that the original data defining function belongs to the smooth class \(\fancyscript{C}^2\), an upper bound for the interpolation error with respect to the \(L_\infty \)-norm is obtained and the uniform convergence of the rational cubic fractal interpolant is deduced. The developed rational fractal interpolation scheme is illustrated with numerical examples and some possible extensions are exposed.KeywordsFractal SplineFractal Interpolation Functions (FIF)Rational SplineFree Shape ParametersIterated Function Systems (IFS)These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.