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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?
Please fill out the following Course Questionnaire at your earliest convenience.
My Frequently Asked Questions on Wavelets (updated on 01/06/2018) 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.
On the occasion of the 120th anniversary of Cornelius Lanczos's birth
the NA group at Manchester made available online a series of video tapes
produced in 1972: http://www.maths.manchester.ac.uk/lanczos. I highly recommend you to watch this video series. In particular, in Tape 1, he talks about the most relevant subjects for this course. Don't miss the video segment around 24-30 minutes where he talks about the importance of Fourier analysis!
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.
K. V. Mardia, 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 Mathematics,
Vol.3, Gordon and Breach Publishers, 1995. Chap. 1 of the Wavelet portion.
Software Package:
A quite good software package ContinuousWavelets.jl written entirely in Julia by my former PhD student, David Weber, is publicly available. Please take a look at it!
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 the following supplementary slides on
analytic signals as well as my talk slides, both of which describe 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 Mathematics, Vol.3,
Gordon and Breach Publishers, 1995. Chap. 2 of the Wavelet portion.
The standard references on the multiresolution approximations are:
J.-P. Kahane and P.-G. Lemarié-Rieusset: Fourier Series
and Wavelets, Studies in the Development of Modern Mathematics, Vol.3,
Gordon and Breach Publishers, 1995. Chap 3 of the Wavelet portion.
WaveletsExt.jl---a Julia extension package to Wavelets.jl---contains functions to compute the auto-correlation shell representation of a given input signal.
Lecture 19: A Library of Orthonormal Bases and Adapted Signal Analysis
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.
The basic references on graph Laplacians and spectral graph theory:
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.
D. Cvetković, P. Rowlinson, and S. Simić: An Introduction to the Theory of Graph Spectra, Vol. 75, London Mathematical Society Student Texts, Cambridge Univ. Press, 2010.
For generalizing the classical Shannon wavelet packets to the graph setting by organizing and grouping the graph Laplacian eigenvectors, see our recent article: