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.
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 Gibbs (or more appropriately Gibbs-Wilbraham) Phenomenon:
Almost all books on Fourier analysis listed above, e.g.,
Dym-McKean, Folland, Pinsky, etc., describe it. Yet the following interesting and
scholastic paper is clearly the best information source:
In higher dimensions, very peculiar things can happen. Gray and Pinsky
discovered a Gibbs-like phenomenon can occur at a point of continuity
in Rd for d > 2.
It is known as the Pinsky phenomenon. See the following articles
for the details.
P. J. Davis and P.
Rabinowitz: Methods of Numerical Integration, Academic Press, 1984 (Reprinted by Dover Pub. Inc., 2007)
Sec. 1.9.
Lecture 6: Functions of Bounded Variations; the Decay Rate of the
Fourier Coefficients II; Fourier Series on Intervals
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.
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.
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!
Lecture 9: From the Sturm-Liouville Theory to Discrete Cosine and Sine Transforms (DCT/DST)
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 and McKean: Sec. 1.7, 1.9.
R. Courant and D. Hilbert: Methods of Mathematical Physics,
Vol. I, First English Edition, John Wiley and Sons, 1953.
Republished as Wiley Classics Library in 1989. See Chap. V
in particular.
For the history and the current status of the Sturm-Liouville theory,
the following book is quite informative:
I also mentioned the Prolate Spheroidal Wave Functions as a part of
the singular Sturm-Liouville problem. There are many references about them,
but perhaps, you may want to check the following recent book and the
references therein:
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.
M. V. Wickerhauser: Adapted Wavelet Analysis from Theory to
Software, A K Peters, Ltd., 1994. Chap. 3.
Lecture 10: Karhunen-Loève Transform (KLT)
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.
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. Ville: "Théorie et applications de la notion de signal
analytique," Cables et Transmissions, 2ème A, no.1,
pp.61-74, 1948. An English version translated by M. D. Godfrey is available at this link.
There have been many recent attempts to adapt Gabor frames for the nonstationary setting, i.e., using Gabor functions with inhomogeneous Heisenberg boxes. See, e.g., the following article and the references therein:
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.
See also the following original papers:
R. Balian: "Un principe d'incertitude fort en théorie du signal ou en mécanique quantique," C. R. Acad. Sci. Paris, vol.292, pp.1357-1362, 1981.
D. E. Vakman: "On the definition of concepts of amplitude, phase, and frequency of a signal," Radio Eng. Electron. Phys., vol.17, no.5, pp.754-759, 1972.
A. Papoulis: Signal Analysis, McGraw-Hill, 1977. Sec.4-2.
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 a variety of efficacy measures for selecting a basis from
basis dictionaries for classification, regression, and other applications,
see, e.g.,
N. Saito: "Local feature extraction and its applications using a library of bases," in Topics in Analysis and Its Applications: Selected Theses (R. Coifman, ed.), pp.269-451, World Scientific, 2000.