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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? Basics of the Fourier Transforms
Please fill out the following Course
Questionnaire and return it to me in person or via email.
G. B. Folland: Real Analysis, 2nd Ed., Wiley
Interscience, 1999. Chap. 8.
Pinsky: Chap.2.
E. M. Stein & G. L. Weiss: Introduction to Fourier
Analysis on Euclidean Spaces, Princeton Univ. Press, 1970. Chap. 1.
Lecture 3: The Heisenberg Uncertainty Principle, Bandlimited
Functions, and Sampling Theorems
Survey on the uncertainty principle (advanced):
G. B. Folland & A. Sitaram: "The uncertainty principle: A
mathematical survey," Journal of Fourier Analysis and
Applications, vol.3, no.3, pp.207-238, 1999.
Sampling Theorems:
R. N. Bracewell: The Fourier
Transform and Its Applications, 2nd Ed., Revised, McGraw-Hill,
1987. Chap. 10.
W. L. Briggs & V. E. Henson: The DFT: An Owner's Manual
for the Discrete Fourier Transform, SIAM, 1995. Sec. 3.4, Chap. 6.
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.
C. E. Shannon: "Communication in the presence of noise," Proc.
IRE, vol.37, pp.10-21, 1949.
P. J. Davis and P.
Rabinowitz: Methods of Numerical
Integration, Academic Press, 1984, Sec. 1.9.
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.
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: Fourier Series on Intervals; Discrete Fourier
Transform
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!
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.
Lecture 10: Karhunen-Loève Expansion
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 KLE 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.
The fascinating book I mentioned in the lecture, containing the
statement on why synthesis is recently getting more important
than analysis, is:
Albert-László Barabási: Linked:
How Everything Is Connected to Everything Else and What It Means for
Business, Science, and Everyday Life, A Plume Book, 2003.
Lectures 14-15: More about Frame Theory; the Balian-Low
Theorem
S. Mallat: A Wavelet Tour of Signal Processing, 2nd
Ed., Academic Press, 1999. Chap. 5.
I. Daubechies: Ten Lectures on Wavelets, SIAM, 1992.
Chap. 3, 4.
J.-P. Kahan 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.
S. Mallat: A Wavelet Tour of Signal Processing, 2nd Ed.,
Academic Press, 1999. Chap. 7.
I. Daubechies: Ten Lectures on Wavelets, SIAM, 1992. Chap.5.
J.-P. Kahan 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.