A Fractal Operator on Some Standard Spaces of Functions

Abstract

By appropriate choices of elements in the underlying iterated function system, methodology of fractal interpolation entitles one to associate a family of continuous selfreferential functions with a prescribed real-valued continuous function on a real compact interval. This procedure elicits what is referred to as α-fractal operator on C(I), the space of all real-valued continuous functions defined on a compact interval I. With an eye towards connecting fractal functions with other branches of mathematics, in this article, we continue to investigate fractal operator in more general spaces such as the space B(I) of all bounded functions and Lebesgue space L p (I), and some standard spaces of smooth functions such as the space C k (I) of k-times continuously differentiable functions, Hölder spaces C k,σ (I), and Sobolev spaces W k,p (I). Using properties of the α-fractal operator, the existence of Schauder basis consisting of self-referential functions is established.

Multivariate Fractal Functions in Some Complete Function Spaces and Fractional Integral of Continuous Fractal Functions

“…In particular, we consider a special class of fractal interpolation functions referred to as the α-fractal function, which has played a considerable role in the theory of univariate fractal approximation. Our work in the current note seeks to show that a few results on the construction of univariate α-fractal functions in various function spaces and associated fractal operator (see, for instance, [1]) carry over to higher dimensions.…”

“…. This section intends to record a few elementary properties of the operator F α ∆,b : X → X, what we call a multivariate selfreferential operator (fractal operator); see also [1]. We shall provide the details only for X = W m,p (Ω), as the other spaces can be similarly dealt with.…”

Multivariate fractal interpolation functions: Some approximation aspects and an associated fractal interpolation operator

“…The aforementioned subclass of FIFs was later named as α-fractal functions to reflect the vectorial parameter α that influences the Hausdorff dimension of the graph of a FIF. Substantial extensions of this theme have been carried out by Navascueés and her coworkers [18,19,20,28]. It is our opinion that the concept of α-fractal functions assisted fractal interpolation to find applications in other fields of mathematics, that belong, in broad sense, to the topic of approximation of functions in various function classes, for instance, in the theory of bases and frames [22,23].…”

Fractal Frames of Functions on the Rectangle

“…If the scale is chosen properly, one can obtain fractal bases of the most used functional spaces, beginning from any basis of these sets. This is done by means of a suitable bounded operator, F α , also known as fractal operator ( [1][2][3][4][5]), transforming systems of ordinary spanning families into their fractal perturbations. In the case of multivariate maps, this operator can no longer be applied to get necessary functions, and some additional tools are required.…”