Construction of fractal bases for spaces of functions
The construction of fractal versions of classical functions as polynomials, trigonometric maps, etc. by means of a particular Iterated Function System of the plane is tackled. The closeness between the classical function and its fractal analogue provides good properties of approximation and interpolation to the latter. This type of methodology opens the use of non-smooth and fractal functions in approximation. The procedure involves the definition of an operator mapping standard functions into their dual fractals. The transformation is linear and bounded and some bounds of its norm are established. Through this operator we define families of fractal functions that generalize the classical Schauder systems of Banach spaces and the orthonormal bases of Hilbert spaces. With an appropriate election of the coefficients of Iterated Function System we define sets of fractal maps that span the most important spaces of functions as C[a, b] or Lp [a, b].
Monotonicity Preserving Rational Cubic Graph-Directed Fractal Interpolation Functions
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 C1-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 C1-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.
A -Fractal Rational Functions and Their Positivity Aspects
Coalescence hidden variable fractal interpolation function (CHFIF) proves more versatile than classical interpolant and fractal interpolation function (FIF). Using rational functions and CHFIF, a general construction of A-fractal rational functions is introduced for the first time in the literature. This construction of A-fractal rational function also allows us to insert shape parameters for positivity-preserving univariate interpolation. The convergence analysis of the proposed scheme is established. With suitably chosen numerical examples and graphs, the effectiveness of the positivity-preserving interpolation scheme is illustrated.
Zipper Rational Quadratic Fractal Interpolation Functions
Inthis article, we propose an interpolation method using a binary parameter called signature such that the graph of the interpolant is an attractor of a suitable zipper rational iterated function system. The presence of scaling factors and signature in the proposed zipper rational quadratic fractal interpolation functions (ZRQFIFs) gives the flexibility to produce a wide variety of interpolants. Using suitable conditions on the scale factor and shape parameter, we construct a C1-continuous ZRQFIF from a C0-continuous ZRQFIF. We also establish the uniform convergence of ZRQFIF to an original data-generating function. Further, we deduce suitable conditions on the IFS parameters and shape parameters to retain the positivity feature associated with a prescribed data by the proposed interpolant.
Constrained data visualization using rational bi-cubic fractal functions
This paper addresses a method to obtain rational cubic fractal functions, which generate surfaces that lie above a plane via blending functions. In particular, the constrained bivariate interpolation discussed herein includes a method to construct fractal interpolation surfaces that preserve positivity inherent in a prescribed data set. The scaling factors and shape parameters involved in fractal boundary curves are constrained suitably such that these fractal boundary curves are above the plane whenever the given interpolation data along the grid lines are above the plane. Our rational cubic spline FIS is above the plane whenever the corresponding fractal boundary curves are above the plane. We illustrate our interpolation scheme with some numerical examples.