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Computational Harmonic Analysis References Page (Winter, 2002)

Course: MAT 280
CRN: 61380
Title: Computational Harmonic Analysis
Class: MW 5:40pm-7:00pm, 693 Kerr

Instructor: Naoki Saito
Office: 675 Kerr
Phone: 754-2121
Email:saito@math.ucdavis.edu
Office Hours: TTh 2:00pm-3:00pm or by appointment via email

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

See Useful Link page

Lecture 2: What is a Signal? Basics of the Fourier Transforms

H. Dym & H. McKean: Fourier Series and Integrals Fourier Series and Wavelets , Academic Press, 1972. Chap. 2.
G. B. Folland: Fourier Analysis and Its Applications , Wadsworth & Brooks/Cole, 1992. Chap. 7.

For quantization, which will not be discussed in this course, see R. M. Gray and D. L. Neuhoff: "Quantization," IEEE Trans. Inform. Theory , vol. 44, no.6, pp.2325-2383, 1998.

Lecture 3: Basics of the Fourier Transforms II: L2 Theory, The Heisenberg Uncertainty Principles
Basics:

Dym & McKean, Chap. 2.
Folland: Fourier Analysis, Chap. 7.
Course handout on the Fourier Inversion Theorem and the L2 Theory

Details of L2 theory:

G. B. Folland: Real Analysis, 2nd Ed., Wiley Interscience, 1999. Chap. 8.
E. M. Stein & G. L. Weiss: Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1970. Chap. 1.

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.

Lecture 4: Bandlimited Functions, Prolate Spheroidal Wave Functions, and Sampling Theorems

Prolate Spheroidal Wave Functions:

D. Slepian: "Some comments on Fourier analysis, uncertainty and modeling," SIAM Review, vol. 25, pp.379-393, 1983.
H. J. Landau: "An overview of time and frequency limiting," in Fourier Techniques and Applications ( J. F. Price, ed.), pp.201-220, Plenum Press, New York, 1985.
D. Slepian: "On bandwidth," Proc. IEEE, vol.63, no.4, pp.292-300, 1976.
D. Slepian & H. O. Pollak: "Prolate spheroidal wave functions, Fourier analysis and uncertainty-I," Bell Syst. Tech. J., vol.40, pp.43-63, 1961.
H. J. Landau & H. O. Pollak: "Prolate spheroidal wave functions, Fourier analysis and uncertainty-II," Bell Syst. Tech. J. , vol.40, pp.65-83, 1961.
H. J. Landau & H. O. Pollak: "Prolate spheroidal wave functions, Fourier analysis and uncertainty-III: The dimension of
the space of essentially time- and band-limited signals," Bell Syst. Tech. J., vol.41, pp.1295-1336, 1962.
D. Slepian: "Prolate spheroidal wave functions, Fourier analysis and uncertainty-IV: Extensions to many
dimensions; generalized prolate spheroidal functions," Bell Syst. Tech. J., vol.43, pp.3009-3057, 1964.
D. Slepian: "Prolate spheroidal wave functions, Fourier analysis and uncertainty-V: The discrete case," Bell Syst. Tech. J., vol.57, pp.1371-1430, 1978.

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 more details on Sampling Theorems and Non-Uniform Sampling Schemes, see:

H. J. Landau: "Sampling, data transmission, and the Nyquist rate," Proc. IEEE, vol.55, no.10, pp.1701-1706, 1967.
A. Aldroubi and K. Gröchenig, "Nonuniform sampling and reconstruction in shift-invariant spaces," SIAM Review, vol.43, no.4, pp.585-620, 2001.

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.

Lecture 5: Fourier Series; Periodization vs Sampling

Basic Fourier Series Theory:

Dym & McKean, Chap. 1.
Folland: Fourier Analysis, Chap. 2.

Generalized Functions (Distributions):

Folland: Fourier Analysis, Chap. 7.
Folland: Real Analysis, Chap. 9.
Stein & Weiss, Chap. 1.
Course handout on the Generalized Functions

For the other stuff I menioned in the class, the details can be found as follows:

Spherical Harmonics:

Dym & McKean, Chap. 4.
Stein & Weiss, Chap. 4.
R. T. Seeley: "Spherical harmonics," Amer. Math. Monthly , vol.73, no.4, part 2, pp.115-121, 1966.

Orthogonal Polynomials:

G. Szegö: Orthogonal Polynomials, 4th Ed., AMS, 1975.
Folland: Fourier Analysis, Chap. 6.

Lecture 6: Discrete Fourier Transform

Course handout on the Discrete Fourier Transform
Briggs & Henson: Chap. 2.
Folland: Fourier Analysis, Sec. 7.6.

For advanced topics on DFT, check out the following articles and book:

J. H. McClellan ad T. W. Parks: "Eigenvalue and eigenvector decomposition of the discrete Fourier transform," IEEE Trans. Audio and Electroacoustics, vol.AU-20, no.1, pp.66-74, 1972 (with comments, vol.AU-21, pp.65, 1973).
L. Auslander and R. Tolimieri: "Is computing with the finite Fourier transform pure or applied mathematics?" Bull. AMS , vol.1, no.6, pp.847-897, 1979.
A. Terras: Fourier Analysis on Finite Groups and Applications , London Mathematical Society Student Texts vol.43, Cambridge Univ. Press, 1999.

Lecture 7: Fast Fourier Transform (FFT)
There are many references on FFT, but the following are particularly useful:

J. W. Cooley and J. W. Tukey: "An algorithm for the machine calculation of complex Fourier series," Math. Comput., vol.19, pp.297-301, 1965.
Briggs & Henson: Chap. 10.
C. F. Van Loan: Computational Frameworks for the Fast Fourier Transform, SIAM, 1992.

For advanced and thought-provoking aspects of FFT, read:

A. Edelman, P. McCorquodale, and S. Toledo: "The future fast Fourier transform?" SIAM J. Sci. Comput., vol.20, no.3, pp.1094-1114, 1999.

The history of the FFT is another very interesting subject. See e.g.,

M. T. Heideman, D. H. Johnson, and C. S. Burrus: "Gauss and the history of the fast Fourier transform," Arch. Hist. Exact Sciences , vol.34, pp.265-277, 1985.

Lecture 8: Basic Sturm-Liouville Theory

Folland: Fourier Analysis: Sec.3.5, 3.6, 7.4.
Dym & McKean: Sec. 1.7, 1.9.

For more details, see e.g.,

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 9: Discrete Cosine & Sine Transforms

N. Ahmed, T. Natarayan, and K. R. Rao: "Discrete cosine transform," IEEE Trans. Comput., vol.COM-23, pp.90-93, 1974.
K. R. Rao and P. Yip: Discrete Cosine Transform: Algorithms, Advantages, and Applications, Academic Press, 1990.
G. Strang: "The discrete cosine transform," SIAM Review, vol.41, no.1, pp.135-147, 1999.
M. V. Wickerhauser: Adapted Wavelet Analysis from Theory to Software, A K Peters, Ltd., 1994. Chap. 3.

Lecture 10: Karhunen-Loè ve 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.
N. Saito: "Image approximation and modeling via least statistically dependent bases, Pattern Recognition, vol.34, no.9, pp.1765-1784, 2001.
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.

Continuous version:

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.
W. D. Ray and R. M. Driver: "Further decomposition of the Karhunen-Loève series representation of a stationary random process," IEEE Trans. Inform. Theory, vol.IT-15, no.6, pp.663-668, 1970.