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Instructor: Naoki Saito Office: 2142 MSB Phone: 754-2121 Email:saito@math.ucdavis.edu Office Hours: MW 3:10pm-4:00pm, or by appointment
The following references are useful and contains much more
details of the topics covered or referred to in my lectures.
I strongly encourage you to take a look at some of them.
Lecture 1: Overture and Motivation;
What is a Signal?
Please fill out the following Course
Questionnaire and return it to me in person or via email.
My
Frequently Asked Questions on Wavelets (updated on 01/04/2010) is available
here. This is an extremely easy and intuitive piece but may be very useful
for further study.
Denoising Enrico Caruso's recording (through the courtesy of Maxim Goldberg) we listened in the class was due to the following paper:
J. Berger, M. Goldberg, & R. Coifman:
"Removing Noise from Music Using Local Trigonometric Bases and Wavelet Packets,"
Journal of the Audio Engineering Society, vol.42, no.10, pp.808-818, 1994.
Lecture 3: Fourier Transforms in L2; The Heisenberg Uncertainty Principle
Details of the L2 theory:
Dym & McKean: Sec. 2.3-2.5.
Pinsky: Sec. 2.4.
G. B. Folland: Real Analysis, 2nd Ed., Wiley
Interscience, 1999. Sec. 8.3.
E. M. Stein & G. L. Weiss: Introduction to Fourier
Analysis on Euclidean Spaces, Princeton Univ. Press, 1970. Sec. 1.2.
Basic references on the Heisenberg inequality/uncertainty principle:
For the historical articles on the sampling theorems, see:
E. T.
Whittaker: "On the functions which are represented by the expansions of
the interpolation-theory," Proc. Royal Soc. Edinburgh, Sec. A,
vol.35, pp.181-194, 1915.
The definition of BV in higher dimensions can be found in:
L. C.
Evans and R. F. Gariepy: Measure Theory and Fine
Properties of
Functions, CRC Press, 1992, Chap.5.
Lecture 6: Functions of Bounded Variations and the Decay Rate of the
Fourier Coefficients II; Fourier Series on Intervals; Discrete Fourier Transform I
Functions of Bounded Variation, the Fourier Coefficients:
C. Lanczos: Discourse on Fourier Series, Hafner
Publishing Co., New York, 1966. Sec 2. This is the best book on 1D
Fourier series from the applied perspective. Unfortunately, this book
is out of print.
V. I. Smirnov: A Course of Higher Mathematics, Vol. V,
Pergamon Press, 1964, Chap. 1. This is out of print too.
Fourier Series on Intervals, Fourier Cosine and Sine Series:
Folland: Fourier Analysis, Sec. 2.4.
C. Lanczos: Applied Analysis, Prentice-Hall, Inc., 1956,
Reprinted by Dover, 1988, Sec. 4.5. This book is still in print. I
strongly urge you to buy this book and read it from cover to cover!
Please read this document carefully. It describes the subtlety of the various
different DFT definitions!
In addition to the references in Lecture 6, I would also recommend:
M. V. Wickerhauser: Adapted Wavelet Analysis from Theory to
Software, A K Peters, Ltd., 1994. Chap. 3.
For the Sturm-Liouville Theory I referred to in today's lecture,
the following are nice references:
Folland: Fourier Analysis: Sec.3.5, 3.6, 7.4.
Dym & McKean: Sec. 1.7, 1.9.
R. Courant & D. Hilbert: Methods of Mathematical Physics,
Vol. I, First English Edition, John Wiley & Sons, 1953.
Republished as Wiley Classics Library in 1989. See Chap. V
in particular.
For the fascinating Laplacian eigenfunctions in higher dimensions, which I
briefly mentioned in today's lecture, see:
My Laplacian eigenfunction resource page containing links to useful talk slides delivered by
the first rate mathematicians and scientists at the workshops and minisymposia I organized over the years.
Discrete version (aka Principal Component Analysis [PCA]):
K. Fukunaga: Introduction to Statistical Pattern Recognition,
2nd Edition, Academic Press, 1990. Chap. 9, Appendix A.
K. V. Madia, J. T. Kent, and J. M. Bibby: Multivariate
Analysis, Academic Press, 1979. Chap. 8.
S. Watanabe: "Karhunen-Loève expansion and factor
analysis: Theoretical remarks and applications," Trans. 4th Prague
Conf. Inform. Theory, Statist. Decision Functions, Random Processes,
Publishing House of the Czechoslovak Academy of Sciences, Prague,
pp.635-660, 1965.
We only discussed the discrete version in the class, but KLT has its
continuous version. The following are some references:
U.
Grenander: Stochastic processes and statistical inference,
Arkiv för Matematik, vol.1, pp.195-277, 1950.
W. B. Davenport and W. L. Root: An Introduction to the Theory of
Random Signals and Noise, McGraw Hill, 1958, republished by IEEE
Press, 1987. Chap. 6.
J.-P. Kahane and P.-G. Lemarié-Rieusset: Fourier Series
and Wavelets, Studies in the Development of Modern Mathematcis,
Vol.3, Gordon and Breach Publishers, 1995. Chap. 1 of the Wavelet portion.
The above references are for analytic signals constructed from an input
signal defined on the entire real axis.
I would strongly recommend to read my following talk slides, in particular,
the section of the analytic signals, which describes those for periodic signals
or signals supported on a finite interval:
J.-P. Kahane and P.-G. Lemarié-Rieusset: Fourier Series
and Wavelets, Studies in the Development of Modern Mathematcis, Vol.3,
Gordon and Breach Publishers, 1995. Chap. 2 of the Wavelet portion.
The standard references on the multiresolution approximations are:
S. Mallat: A Wavelet Tour of Signal Processing, 3rd Ed.,
Academic Press, 2009. Chap. 7.
I. Daubechies: Ten Lectures on Wavelets, SIAM, 1992. Chap. 5.
J.-P. Kahane and P.-G. Lemarié-Rieusset: Fourier Series
and Wavelets, Studies in the Development of Modern Mathematcis, Vol.3,
Gordon and Breach Publishers, 1995. Chap 3 of the Wavelet portion.
For the asymptotic convergence of the cardinal B-spline to the Gaussian,
see
For more about the orthonormal wavelet bases, see e.g.,
S. Mallat: A Wavelet Tour of Signal Processing, 3rd Ed.,
Academic Press, 2009. Chap. 7.
I. Daubechies: Ten Lectures on Wavelets, SIAM, 1992. Chap. 7.
For the wavelet packets and their dual, i.e., local cosines, see e.g.,
S. Mallat: A Wavelet Tour of Signal Processing, 3rd Ed.,
Academic Press, 2009. Chap. 8.
Finally, for harmonic analysis on graphs or wavelets on graphs,
see e.g.,
F. R. K. Chung: Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, no. 92, Amer. Math. Soc., 1997.
Some of the chapters are available from her website.