Uniform Continuity of Fractal Interpolation Function

Xuezai Pan,Minggang Wang,Xudong Shang,Jixiang Yang

doi:10.1155/2020/7840432

Abstract

In order to research analysis properties of fractal interpolation function generated by the iterated function system defined by affine transformation, the continuity of fractal interpolation function is proved by the continuous definition of function and the uniform continuity of fractal interpolation function is proved by the definition of uniform continuity and compactness theorem of sequence of numbers or finite covering theorem in this paper. The result shows that the fractal interpolation function is uniformly continuous in a closed interval which is from the abscissa of the first interpolation point to that of the last one.

Highlights

In 1960s, fractal geometry was regarded as a new interdiscipline firstly discovered by American mathematician Mandlebrot [1,2,3,4,5]
Because most extremely irregular graphics in nature and very irregular social phenomenon are researched in the fractal geometry field, fractal geometry is called natural geometry. erefore, fractal geometry is applied in almost all fields, such as mathematics, physics, chemistry, engineering, social science, and art [6,7,8,9]
The fractal interpolation function method has been paid more and more attention by mathematicians. e theory of fractal interpolation function generated by the iterated function system defined by affine transformation was firstly proposed by Barnsley [14,15,16] and Massopust [17, 18]. ey found that any part of a fractal graphic is similar to the whole, so they used mathematical language to express the similar iterated process. at is to say, first, the iterated function system consisting of affine transformation is defined and it is proved that the iterated function system has a unique attractor that is the fixed point

Summary

Introduction

In 1960s, fractal geometry was regarded as a new interdiscipline firstly discovered by American mathematician Mandlebrot [1,2,3,4,5]. Because most extremely irregular graphics in nature and very irregular social phenomenon are researched in the fractal geometry field, fractal geometry is called natural geometry. The fractal interpolation function method has been paid more and more attention by mathematicians. E theory of fractal interpolation function generated by the iterated function system defined by affine transformation was firstly proposed by Barnsley [14,15,16] and Massopust [17, 18]. The dimension theory and integrability of fractal interpolation function have been studied by Barnsley and Massopust. Based on the research of fractal interpolation function above, the continuity and uniform continuity of the fractal interpolation function-generated iterated function systemdefined affine mapping are proved in the paper

Main Concepts and Lemmas

The Continuity and the Uniform Continuity of Fractal Interpolation Function

Conclusion