Wavelets in Physics

Recent advances have shown wavelets to be an effective, and even necessary, mathematical tool for theoretical physics. This book is a timely overview of the progress of this new frontier. It includes an introduction to wavelet analysis, and applications in the fields of high energy physics, astrophysics, cosmology and statistical physics. The topics are selected for the interests of physicists and graduate students of theoretical studies. It emphasizes the need for wavelets in describing and revealing structure in physical problems, which is not easily accomplishing by other methods.

Preface

v

(2)

List of Contributors

vii

Peter Carruthers' Notes on Wavelet Analysis

1

(14)

A. Wavelet Analysis of Music, Art and Poetry

1

(2)

B. Moment and Wavelet Analysis of Correlations in Multihadron and Galaxy Distribution

3

(12)

An Introduction of the Wavelet Transform

15

(32)

J. Pando

L.-Z. Fang

1. The Predecessors of Wavelet Transform

15

(3)

2. The Wavelet Transform Basis Functions

18

(4)

3. The Continuous Wavelet Transform

22

(2)

4. General Properties of the Wavelet Transform

24

(3)

5. An Example of the Discrete Wavelet: The Haar Wavelet

27

(4)

6. Multi-resolution Analysis

31

(2)

7. Compactly Supported Orthogonal Basis

33

(5)

8. Relationship between Fourier and DWT Expansion

38

(2)

9. n-Dimensional Wavelets

40

(1)

10. Summary

40

(2)

Appendix

42

(2)

A. Relationship between Fourier Coefficients and FFCs

42

(1)

B. Parseval Theorem of the DWT

43

(1)

References

44

(3)

Wavelet Applications in High Energy and Nuclear Physics

47

(42)

R.L. Thews

1. Introduction

47

(1)

2. Disordered Chiral Condensate (DCC)

47

(3)

3. Wavelet Formalism

50

(10)

4. Analysis of Events Generated in Simulations Based on the Linear Sigma Model

60

(19)

5. Analysis of Experimental Data

79

(5)

6. Conclusions and Outlook

84

(2)

References

86

(3)

Correlation Structure of Wavelet Cascades

89

(44)

M. Greiner

J. Giesemann

1. Introduction

89

(2)

2. Some Basics about Wavelets

91

(8)

3. Multiplicative Haar-Wavelet Cascade

99

(21)

4. Multiplicative Daubechies-Wavelet Cascade

120

(8)

5. Conclusion

128

(2)

References

130

(3)

The Discrete Wavelet Transform in Cosmology: Power Spectrum and Multiscale Identification

133

(34)

L.-Z. Fang

J. Pando

1. Introduction

133

(3)

2. The Discrete Wavelet Transform Power Spectrum

136

(7)

3. Spectra Reconstruction

143

(6)

4. The Power Spectrum of the Ly-alpha Forests

149

(4)

5. Multi-Scale Identification of Clusters (1-D)

153

(7)

6. Multi-Scale Identification of Clusters (3-D)

160

(3)

7. Conclusion

163

(1)

References

164

(3)

The Discrete Wavelet Transform in Cosmology: Quasilinear and Non-linear Evolutions

167

(46)

L.-Z. Fang

J. Pando

1. Introduction

167

(2)

2. Higher Order Statistics via the Wavelet Coefficients

169

(9)

3. Scale-Scale Correlations

178

(4)

4. Detection of Non-Gaussianity of Quasilinear Regime

182

(7)

5. Scaling of Gravitational Clustering

189

(7)

6. Bias of Galaxy Distributions

196

(11)

7. Conclusion

207

(1)

8. Epilogue and Outlook

208

(1)

References

208

(5)

Index

213

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