Fractal rational functions and their approximation properties

A Fractal Operator Associated with Bivariate Fractal Interpolation Functions on Rectangular Grids

“…If b = Lf , where L is a bounded linear operator, then the operator F α ∆,L is a bounded linear operator termed the fractal operator. This operator formulation of fractal functions somewhat hidden in the construction of FIFs enables them to interact with other traditional branches of mathematics including operator theory, complex analysis, harmonic analysis and approximation theory [18,19,20,21,22,26]. More recently, the second author and collaborators identified suitable values of the parameters so that the α-fractal function f α ∆,b preserves the shape properties inherent in the source function f [27].…”

Approximation of rough functions

“…Each function h α in this family is referred to as α-fractal function or "fractal perturbation" corresponding to h. In our present setting, the function f is the fractal perturbation corresponding to g 0 + f 0 with base function f 0 and constant scale vector α whose components are a. Therefore, the α-fractal function and the approximation classes obtained through the corresponding fractal operator (see, for instance, [19,23]) can also be discussed using the present formalism.…”

Fractal Bases for Banach Spaces of Smooth functions

“…This operator connects fractal functions with fields such as functional analysis, operator theory and approximation theory (see, for instance, [8,9,16]). In [17], suitable values of the scaling factors are identified so that f α preserves fundamental shape properties such as positivity, monotonicity and convexity inherent in the function f .…”