**Author**: Carlos Enrique D'Attellis,Elena M. Fernandez-Berdaguer

**Publisher:** Springer Science & Business Media

**ISBN:** 9780817639532

**Category:** Mathematics

**Page:** 345

**View:** 8690

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This book contains 14 invited chapters addressing applications and interactions between wavelet theory and scientific, medical and geophysical problems. Topics covered include EGG signals, spectral analysis, wavelet transform from orthogonal spline wavelets, numerical modeling of Maxwell's equation, wavelet networks and nonlinear processes, etc. It is addressed to an interdisciplinary readership of professional applied mathematicians, electrical engineers, physicists and other scientists especially interested in applying the new ideas and techniques. Series: Applied and Numerical Harmonic Analysis (ANHA) Contents I. Theory and Implementations 1. Singular integrals related to the Monge-Ampere equation -L.A.Caffarelli and C. Gutierrez 1. Introduction 2. The maximal function 3. Application to singular integral operators 4. References 2. Wavelet characterization of functions with conditions on the mean oscillation -H. Aimar and A. Bernardis 1. Introduction 2. Wavelet Bases 3. Functions with conditions on the mean oscillation: MO spaces 4. Sequential spaces of Carleson type: C spaces 5. C as a necessary condition for the wavelet coefficients of MO functions 5.1 The Daubechies wavelet case 5.2 The Meyer wavelet case 6. C as a sufficient condition for MO : The finite case 6.1 The Daubechies wavelet case 6.2 The Meyer wavelet case 7. C as sufficient condition for MO : The general case 8. Acknowledgments 9. References 3. Undecimated Wavelet Transform from Orthogonal Spline Wavelets -E.P. Serrano and M.A. Fabio 1. Introduction 2. The Undecimated Discrete Wavelet Transform 3. Undecimate Cardinal Algorithm 4. Properties of the Undecimated Discrete Transform 5. Extended Multiresolution Analysis for V 0 6. Decomposition of a Signal in V 0 7. Conclusion 8. References 4. Oblique Multiwavelet Bases -A. Aldroubi 1. Introduction 1.1 An oblique wavelet based on the Haar multiresolution 2. Multiscaling Functions and Multiwavelet Bases 2.1 Multiscaling functions 2.2 Multiwavelets 2.3 Classification of wavelet bases 2.4 Two-scale equations 3. Construction of Oblique Wavelet and Multiwavelet Bases 3.1 Oblique multiwavelet bases based on the Hermite cubic splines MRA 4. Fast Filter-Bank Algorithms 4.1 Initialization 5. References 5. Frames and Riesz bases: a short survey -S.J. Favier and R.A. Zalik 1. Introduction 2. Frames and Riesz bases in Hilbert Spaces 3. Frame and Basis Perturbations 4. Special Classes of Frames and The Proyection Method 5. Exponential Frames and Bases 6. Wavelet Frames, Bases, and Bessel Sequences 7. References 6. Fourier Analysis of Petrov-Galerkin Methods Based on Biorthogonal Multiresolution Analyses -S.M. Gomes and E. Cortina 1. Introduction 2. Notation and some definitions 3. A Petrov-Galerkin method for the KdV equation 4. The linear case: Fourier analysis 4.1 Convergence results 4.2 Conditions for stability 5. Biorthogonal framework 5.1 Spline biorthogonal scaling functions 5.2 Algorithms for the calculation of a(k), b(l,k) and c(k) 6. Numerical results 7. Acknowledgment 8. References II. Applications to Biomedical Sciences 7. Fine Structure of ECG Signal using Wavelet Transform -H. Rix and O. Meste 1. Introduction 2. Localization of Late Potentials by Time Frequency Representations 3. Signal Shape Differences 3.1 Shape classifications of 1-D signals 3.2 Shape classification of 2-D signals 4. Conclusion 5. References 8. Spectral Analysis of Cardiorespiratory Signals -M. Risk, J. Sobh, R. Armentano, A. Ramirez and P. Saul 1. Introduction 2. Short term variability 2.1 Spectral Analysis Methods 2.2 Spectral Analysis 2.3 Autoregressive and moving average modeling 2.4 Fast Fourier Transform method 2.5 Blackman-Tukey method 2.6 Transfer Function 3. Long term variability 3.1 Time domain methods 3.2 Frequency domain methods 4. Discussion 5. References 9. Characterization of Epileptic EEG Time Series (I): Gabor Transform and Nonlinear Dynamics Methods -S. Blanco, S. Kochen, R. Quian Quiroga, L. Riquelme, O. Rosso and P. Salgado 1. Introduction 2. Experimental Setup and Clinical Data 3. Armonic Analysis of EEG Data 4. Time-Frequency Analysis Based on Gabor Transform 4.1 Analysis of EEG Signal I 4.2 Analysis of EEG Signal II 4.3 Information Transfer Analysis 5. Nonlinear Dynamics Analysis 5.1 Stationarity 5.2 Dynamical Systems 5.3 Attractors 5.4 Attractor Reconstruction 5.5 Choosing the Optimal Time Delay 5.6 Choosing the Minimum Embedding Dimension 5.7 Correlation Dimension 5.8 Lyapunov Exponent 5.9 Analysis of Signal II 6. Final Remarks 7. Acknowledgements 8. References 10. Characterization of Epileptic EEG Time Series (II): Wavelet Transform and Information Theory -C. D'Attellis, L. Gamero, S. Isaacson, R. Sirne and M. Torres 1. Introduction 2. Data Collection 3. Theoretical Background 3.1 Wavelet Analysis and Filters 3.2 Entropy 4. Method I: Energy Based Detection Algorithm 4.1 Results and Comparisons 4.2 Discussion 5. Method II: Multiresolution Entropy 5.1 Results 5.2 Discussion 6. Conclusions 7. References III. Applications in Physical Sciences 11. Wavelet Networks for Modelling Nonlinear Processes -N. Roqueiro and E.L. Lima 1. Introduction 2. Models for non-linear system identification 2.1 Volterra's series 2.2 Block oriented models 2.3 Non-linear models linear in the parameters 3. Wavelet networks 3.1 Multivariable wavelet networks 3.2 Parallel structure of single variable wavelets 3.3 Conclusions 4. Examples 4.1 The non-linear identification problem 4.2 Identification of a chaotic attractor 4.3 Continuous stirred tank reactor identification 4.4 Conclusions 5. General conclusion 6. References 12. Higher order asymptotic boundary conditions for an oxide region in a semiconductor device -I. Gamba 1. Introduction 2. The full problem 2.1 Regularity of the full problem 2.2 Equivalent problem formulated in terms of the Fourier representation 2.3 Higher order asymptotic behavior of the Oxide boundary condition 2.4 Conclusions 3. References 13. Estimation of the complex plain-wave modulus in viscoelastic media -E.M. Fernandez-Berdaguer and J.E. Santos 1. Introduction 1.1 Description of the Viscoelastic Model 1.2 Formulation of the Estimation Problem 1.3 The Gateaux Derivatives 1.4 The Algorithms 1.5 The Adjoint Problem 1.6 Numerical Experiments 2. References 14. Numerical Modelling of Maxwell's Equations with Applications to Magnetotellurics -J.E. Santos and L. Guarracino 1. The Differential Model and the Interative Hybrid Finite Element Domain Decomposition Algorithm 1.1 The Differential Model 1.2 A Differential Domain Decomposition Formulation 1.3 The Iterative Hybrid Finite Element Domain Decomposition Procedure 1.4 Experimental Calculations 1.5 Some remarks on the implementation of the algorithm 2. Conclusions 3. References