Fractal Frames of Functions on the Rectangle

María A Navascués,Ram Mohapatra,Md Nasim Akhtar

doi:10.3390/fractalfract5020042

María A Navascués, Ram Mohapatra + Show 1 more

Open Access

https://doi.org/10.3390/fractalfract5020042

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Abstract

In this paper, we define fractal bases and fractal frames of L2(I×J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L2(I×J) with the tensor product space L2(I)⨂L2(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L2(I×J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable.

Highlights

In a rapidly changing world, with unexpected outcomes, the scientific community has to make particular effort to provide a deeper knowledge and understanding of the reality and natural environment surrounding us
While it is true that fractal approximation is an active field of research currently, and there is an abundant bibliography about multivariate fractal interpolation functions, our approach has some specificities
The factors are of type α-fractal functions, which constitute a generalization of any map defined on a compact real interval

Summary

Introduction

In a rapidly changing world, with unexpected outcomes, the scientific community has to make particular effort to provide a deeper knowledge and understanding of the reality and natural environment surrounding us. The first one is that the functional bases proposed are a generalization of any product basis (classical or not) This fact provides a wide spectrum of maps, in order to choose the optimum for a particular application, extending the analytical, geometric and dynamical possibilities. The mappings presented own all the advantages of the traditional functions because they include them as particular cases (taking the scale vectors equal to zero) They provide new non-smooth geometric objects to model complex behaviors. We define fractal bases and fractal frames of L2 ( I × J ), in order to approximate two-dimensional square-integrable maps whose domain is a rectangle This is accomplished by means of the identification of L2 ( I × J ) with the tensor product space. Due to the last item, we can consider that the fractal maps f α are generalizations of any function

Fractal Frames on the Rectangle

Conclusions