Monotonicity Preserving Rational Cubic Graph-Directed Fractal Interpolation Functions.

TLDR: This work proposes a new class \(\mathscr {C}^1\)-rational cubic graph-directed FIFs (RCGDFIFs) using cubic rational function involving two shape parameters in each sub-interval and deduced sufficient condition based on the restriction of the corresponding rational GDIFS parameters.

Abstract: The idea of graph-directed fractal interpolation function (FIF) is introduced recently to represent several dependent data sets from graph-directed iterated function system (GDIFS). When dependent data sets are generated from \(\mathscr {C}^1\)-smooth functions with irregular derivatives, it is not ideal to use affine FIF or classical splines in such scenario. Thus, we initiate the use of smooth graph directed FIFs for two or more sets of interpolation data that are not independent, and generated from original smooth functions having fractal characteristics in their derivatives. For this task, we have proposed a new class \(\mathscr {C}^1\)-rational cubic graph-directed FIFs (RCGDFIFs) using cubic rational function involving two shape parameters in each sub-interval. For applications of the proposed RCGDFIFs in modeling of monotonic data sets, we have deduced sufficient condition based on the restriction of the corresponding rational GDIFS parameters.