Frontiers of Fractal Analysis

Recent Advances and Challenges

Santo Banerjee, A. Gowrisankar, Santo Banerjee, A. Gowrisankar

• Publisher: CRC Press
• Year: 2022
• Print ISBN: 9781032138671
• eBook ISBN: 9781000625950
• Edition: 1
• Topic: Mathematics
• Subtopic: Arithmetic
• Index: Mathematics

The history of describing natural objects using geometry is as old as the advent of science itself, in which traditional shapes are the basis of our intuitive understanding of geometry. However, nature is not restricted to such Euclidean objects which are only characterized typically by integer dimensions. Hence, the conventional geometric approach cannot meet the requirements of solving or analysing nonlinear problems which are related with natural phenomena, therefore, the fractal theory has been born, which aims to understand complexity and provide an innovative way to recognize irregularity and complex systems. Although the concepts of fractal geometry have found wide applications in many forefront areas of science, engineering and societal issues, they also have interesting implications of a more practical nature for the older classical areas of science. Since its discovery, there has been a surge of research activities in using this powerful concept in almost every branch of scientific disciplines to gain deep insights into many unresolved problems.

This book includes eight chapters which focus on gathering cutting-edge research and proposing application of fractals features in both traditional scientific disciplines and in applied fields.

Chapter 1 Some Remarks on Multivariate Fractal Approximation

Megha Pandey*, Vishal Agrawal and Tanmoy Som

Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, India 221005.

Emails: [email protected]; [email protected]

* Corresponding author: [email protected]

1.1 Introduction

The number π is well known in science, it is listed as ratio of a circle’s circumference to its diameter. But one might wonder what is the exact value of π. While solving a numerical problem, we use 3.14 or 3.141, etc., in place of π. But all these values are the nearest possible value of its exact value, these values are known as approximate values. In fact there exist a lot of mathematical numbers whose exact value is a hard nut to crack so we use an approximate value instead of exact value. Approximation is used not only for numbers but we can use it for functions also, i.e., we can even ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Preface
• Table of Contents
• List of Contributors
• 1. Some Remarks on Multivariate Fractal Approximation
• 2. Fractal Interpolation: From Global to Local, to Nonstationary and Quaternionic
• 3. A Study on Fractal Operator Corresponding to Non-stationary Fractal Interpolation Functions
• 4. Fractal Calculus
• 5. Perspective of Fractal Calculus on Types of Fractal Interpolation Functions
• 6. On the Borel Regularity of the Relative Centered Multifractal Measures
• 7. A Mixed Multifractal Analysis of Vector-valued Measures: Review and Extension to Densities and Regularities of Non-necessary Gibbs Cases
• 8. Multifractal Dimensions and Fractional Differentiation in Automated Edge Detection on Intuitionistic Fuzzy Enhanced Image
• Index