This book introduces the fractal interpolation functions (FIFs) in approximation theory to the readers and the concerned researchers in advanced level. FIFs can be used to precisely reconstruct the naturally occurring functions when compared with the classical interpolants.
The book focuses on the construction of fractals in metric space through various iterated function systems. It begins by providing the Mathematical background behind the fractal interpolation functions with its graphical representations and then introduces the fractional integral and fractional derivative on fractal functions in various scenarios. Further, the existence of the fractal interpolation function with the countable iterated function system is demonstrated by taking suitable monotone and bounded sequences. It also covers the dimension of fractal functions and investigates the relationship between the fractal dimension and the fractional order of fractal interpolation functions. Moreover, this book explores the idea of fractal interpolation in the reconstruction scheme of illustrative waveforms and discusses the problems of identification of the characterizing parameters.
In the application section, this research compendium addresses the signal processing and its Mathematical methodologies. A wavelet-based denoising method for the recovery of electroencephalogram (EEG) signals contaminated by nonstationary noises is presented, and the author investigates the recognition of healthy, epileptic EEG and cardiac ECG signals using multifractal measures.
This book is intended for professionals in the field of Mathematics, Physics and Computer Science, helping them broaden their understanding of fractal functions and dimensions, while also providing the illustrative experimental applications for researchers in biomedicine and neuroscience.
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1 Mathematical Background of Deterministic Fractals |
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1 | (20) |
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1 | (4) |
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1.2 Iterated Function System |
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5 | (4) |
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1.3 Countable Iterated Function System |
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9 | (2) |
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1.4 Local Countable Iterated Function System |
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11 | (4) |
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1.4.1 Local Iterated Function System |
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11 | (1) |
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1.4.2 Existence and Analytical Properties of LCIFS |
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12 | (3) |
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15 | (2) |
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1.6 Generalized Fractal Dimensions |
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17 | (2) |
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18 | (1) |
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1.6.2 Limiting Cases of Generalized Fractal Dimensions |
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19 | (1) |
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1.6.3 Range of Generalized Fractal Dimensions |
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19 | (1) |
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19 | (2) |
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21 | (16) |
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21 | (1) |
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2.2 Interpolation Functions |
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22 | (1) |
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2.3 Fractal Interpolation Function |
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23 | (6) |
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2.4 Hidden Variable Fractal Interpolation Function |
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29 | (3) |
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2.5 Classical Calculus on Fractal Interpolation Functions |
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32 | (3) |
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35 | (2) |
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3 Fractional Calculus on Fractal Functions |
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37 | (24) |
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37 | (2) |
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3.2 Linear Fractal Interpolation Function |
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39 | (10) |
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3.3 Riemann-Liouville Fractional Calculus Quadratic FEF |
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49 | (4) |
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53 | (1) |
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3.5 Fractional Calculus of Quadratic FIF with Variable Scaling Factors |
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54 | (5) |
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59 | (2) |
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4 Fractal Interpolation Function for Countable Data |
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61 | (18) |
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61 | (1) |
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4.2 Existence of FIF for Countable Data Set |
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62 | (6) |
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4.3 Fractional Calculus on Interpolation Function of Sequence of Data |
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68 | (9) |
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77 | (2) |
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5 Multifractal Analysis and Wavelet Decomposition in EEG Signal Classification |
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79 | (40) |
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79 | (1) |
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79 | (4) |
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5.2.1 Synthetic Weierstrass Signals |
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79 | (2) |
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5.2.2 Clinical EEG Signals |
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81 | (2) |
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83 | (1) |
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83 | (1) |
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83 | (1) |
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5.3.3 Normal Probability Plot |
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83 | (1) |
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5.4 Development of Multifractal Analysis in EEG Signal Classification |
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83 | (12) |
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5.4.1 Modified Generalized Fractal Dimensions |
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84 | (1) |
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5.4.2 Improved Generalized Fractal Dimensions |
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85 | (2) |
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5.4.3 Advanced Generalized Fractal Dimensions |
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87 | (1) |
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5.4.4 Methods to Analyze the Fractal Time Signals |
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88 | (1) |
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5.4.5 Results and Discussions |
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89 | (6) |
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5.5 Multifractal-Wavelet Based Denoising in EEG Signal Classification |
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95 | (22) |
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5.5.1 Discrete Wavelet Transform |
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96 | (3) |
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5.5.2 Wavelet Denoising of Signals |
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99 | (1) |
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5.5.3 Results and Discussions |
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100 | (17) |
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117 | (2) |
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6 Fuzzy Multifractal Analysis in ECG Signal Classification |
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119 | (10) |
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119 | (1) |
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6.2 Fuzzy Multifractal Analysis for Fractal Signals |
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119 | (2) |
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6.2.1 Fuzzy Renyi Entropy |
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119 | (1) |
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6.2.2 Fuzzy Generalized Fractal Dimensions for Signals |
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120 | (1) |
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6.3 Fuzzy Generalized Fractal Dimensions for Deterministic Fractal Waveforms |
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121 | (3) |
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6.4 Experimental ECG Data |
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124 | (1) |
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6.5 Fuzzy Generalized Fractal Dimensions for Clinical ECG Signals |
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125 | (2) |
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127 | (2) |
| References |
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129 | |
Santo Banerjee was working as Associate Professor, in the Institute for Mathematical Research (INSPEM), University Putra Malaysia, Malaysia till 2020, and also a founder member of the Malaysia-Italy Centre of Excellence in Mathematical Science, UPM, Malaysia. He is now associated with the Department of Mathematics, Politecnico di Torino, Italy. His research is mainly concerned with Nonlinear Dynamics, Chaos, Complexity and Secure Communication. He is a Managing Editor of EPJ Plus (Springer)
D. Easwaramoorthy received the M.Sc. Degree in Mathematics from the Bharathidasan University, Tiruchirappalli, India in 2008 and the Ph.D. Degree in Mathematics from The Gandhigram Rural Institute (Deemed to be University), Gandhigram, Dindigul, India in 2013. Currently, he is an Assistant Professor (Senior) in the Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (Deemed to be University), Vellore, Tamil Nadu, India. His research interests include Fractal Analysis, Fuzzy Metric Spaces, Fuzzy Mathematics, Signal & Image Analysis; and Biomedical Signal Analysis. A. Gowrisankar received the M.Sc. and Ph.D. Degrees in Mathematics from the The Gandhigram Rural Institute (Deemed to be University), Gandhigram, Dindigul, India in 2012 and 2017 respectively. He got institute postdoctoral fellowship from Indian Institute of Technology Guwahati (IITG), Guwahati, Assam, India in 2017. At present, he is an Assistant Professor in the Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (Deemed to be University), Vellore, Tamil Nadu, India. His broad area of research includes Fractal Analysis, Image Processing, Fractional Calculus and Fractal Functions.
