Wavelets: Mathematics and applications

The notion of wavelets is defined. It is briefly described what wavelets are, how to use them, when we do need them, why they are preferred, and where they have been applied. Then one proceeds to the multiresolution analysis and fast wavelet transform as a standard procedure for dealing with discrete wavelets. It is shown what specific features of signals (functions) can be revealed by this analysis, but cannot be found by other methods (e.g., by the Fourier expansion). Finally, some examples of practical application are given. Rigorous proofs of mathematical statements are omitted, and the reader is referred to the corresponding literature.

[1]  A. Haar.  Zur Theorie der orthogonalen Funktionensysteme , 1910 .

[2]  L. Amaral,et al.  Multifractality in human heartbeat dynamics , 1998, Nature.

[3]  Gerald Kaiser,et al.  A Friendly Guide to Wavelets , 1994 .

[4]  T. Koornwinder.  Wavelets : an elementary treatment of theory and applications , 1993 .

[5]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[6]  Yves Meyer,et al.  Progress in wavelet analysis and applications , 1993 .

[7]  S. T. Buckland,et al.  An Introduction to the Bootstrap. , 1994 .

[8]  V. A. Nechitailo,et al.  Wavelets and their uses , 2001 .

[9]  Charles K. Chui,et al.  An Introduction to Wavelets , 1992 .

[10]  Igor Dremin,et al.  Precursors of stall and surge processes in gas turbines revealed by wavelet analysis , 2002 .

[11]  V. A. Nechitailo,et al.  Wavelet patterns in nucleus–nucleus collisions at 158A GeV , 2000 .

[12]  S. Thurner,et al.  Multiresolution Wavelet Analysis of Heartbeat Intervals Discriminates Healthy Patients from Those with Cardiac Pathology , 1997, adap-org/9711003.

[13]  Y. Meyer.  Wavelets and Operators , 1993 .

[14]  N. M. Astaf’eva.  Wavelet analysis: basic theory and some applications , 1996 .

[15]  I. Daubechies.  Ten Lectures on Wavelets , 1992 .

[16]  J. Morlet,et al.  Wave propagation and sampling theory—Part I: Complex signal and scattering in multilayered media , 1982 .

[17]  Wei Lin,et al.  Wavelet Analysis and Applications , 2011 .

[18]  H E Stanley,et al.  Scale-independent measures and pathologic cardiac dynamics. , 1998, Physical review letters.

[19]  J. Berg,et al.  Wavelets in Physics , 2004 .

[20]  Bruno Torrésani,et al.  Practical Time-Frequency Analysis , 1998 .

[21]  G. Weiss,et al.  A First Course on Wavelets , 1996 .

[22]  M. Y. Hussaini,et al.  Wavelets: theory and applications , 1996 .

[23]  Y. Meyer,et al.  Wavelets: Calderón-Zygmund and Multilinear Operators , 1997 .

[24]  A. Aldroubi,et al.  Wavelets in Medicine and Biology , 1997 .

[25]  I. Dremin,et al.  SCALING LAWS FOR DENSITY CORRELATIONS AND FLUCTUATIONS IN MULTIPARTICLE DYNAMICS , 1995, hep-ph/9508325.

[26]  I. Dremin,et al.  Pattern Recognition in High Multiplicity Events , 1997, hep-ex/9705003.

[27]  M. Victor Wickerhauser,et al.  Wavelets: Algorithms and Applications (Yves Meyer) , 1994, SIAM Rev..