
NWC Courses
The Norbert Wiener Center prides itself on providing its graduate students with stateofthe art education in mathematics and applied mathematics. Faculty at the NWC teach courses ranging from the rudimentary analysis courses required for graduate qualifying exams, to the highlyfocused topics courses.
Below is a list of recent courses taught by Norbert Wiener Center Faculty.
Fall 2019

MATH464  Transform Methods for Scientists and Engineers Radu Balan
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Fourier transform, Fourier series, discrete fast Fourier transform (DFT and FFT).
Laplace transform. Poisson summations, and sampling.
Optional Topics: Distributions and operational calculus, PDEs,
Wavelet transform, Radon transform and applications such as Imaging,
Speech Processing, PDEs of Mathematical Physics, Communications, Inverse Problems.

MATH858R/AMSC808R  Harmonic Analysis Methods for Random and LowRank Matrices Radu Balan
Course Page

MATH634  Harmonic Analysis John J. Benedetto
Syllabus
1. Fourier analysis on Euclidean spaces including Fourier series
2. Distribution theory and applications
3. Fourier analysis on locally compact abelian groups
4. The uncertainty principle
5. Modern applications of Fourier analysis
6. Discrete Fourier series (DFT) and the Fast Fourier Transform (FFT)
5. Sampling theory and the relations between Fourier transforms,
Fourier series, and DFTs
6. Background for timefrequency analysis, wavelet theory on
Euclidean spaces and local fields, image processing, dimension reduction,
compressive sensing, Wiener's Generalized Harmonic Analysis, and frames

MATH630  Real Analysis I Wojtek Czaja
Lebesgue measure and the Lebesgue integral on R, differentiation of functions of bounded variation, absolute continuity and fundamental theorem of calculus, Lp spaces on R, RieszFischer theorem, bounded linear functionals on Lp, measure and outer measure, Fubini's theorem.
Spring 2019

MATH416  Applied Harmonic Analysis: An Introduction to Signal Processing Wojtek Czaja
MATH 416 introduces the mathematical concepts arising in signal analysis from the applied harmonic analysis point of view. Topics include: applied linear algebra, Fourier series, discrete Fourier transform, Fourier transform, Shannon Sampling Theorem, Wavelet transform, wavelet bases, multiresolution analysis, and discrete wavelet transform.

MATH631  Real Analysis II Radu Balan
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Real Analysis II is the continuation of MATH 630 Real Analysis I. In particular it presents: Abstract measure and integration theory, metric spaces, Baire category theorem and uniform boundedness principle, RadonNikodym theorem, Riesz Representation theorem, Lebesgue decomposition, Banach and Hilbert Spaces, BanachSteinhaus theorem, topological spaces, ArzelaAscoli and StoneWeierstrass theorems, compact sets and Tychonoff's theorem.
Fall 2018

MATH858W  Frames, Sampling, and Wavelets John J. Benedetto
Syllabus
1. Timefrequency (Gabor) analysis on R^d and the role of the Heisenberg group.
The short time Fourier transform (STFT), narrow band ambiguity function, and
applications.
2. Wavelet theory on R^d and the role of the ax+b group. The wide band ambiguity
function and applications.
3. Frames. Timefrequency (Gabor) and wavelet frames, frame multiresolution analysis,
Grassmannian frames, harmonic and group frames, and the role of the DFT. Applications
to quantum information theory and open problems.
4. Sampling theory. Uniform sampling on R and R^d, Poisson summation and
applications, quasicrystals and nonuniform Poisson summation, balayage and nonuniform
sampling, SigmaDelta quantization and nonlinear sampling, dynamical
sampling.
5. Compressed sensing (sampling). Gabor and wavelet matrix equations, sparse
solutions, the role of the Donoho/Stark and Tao uncertainty principles for compressed
sensing, mathematical properties of Gabor matrices for CAZAC generating
functions.
6. Uncertainty principles. The BalianLow phenomenon and Bourgain's theorem.

MATH632  Functional Analysis Radu Balan
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Abstract measure and integration theory, metric spaces, Baire category theorem and uniform boundedness principle, RadonNikodym theorem, Riesz Representation theorem, Lebesgue decomposition, Banach and Hilbert Spaces, BanachSteinhaus theorem, topological spaces, ArzelaAscoli and StoneWeierstrass theorems, compact sets and Tychonoff's theorem.
Spring 2018

MATH631  Real Analysis II Radu Balan
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Abstract measure and integration theory, metric spaces, Baire category theorem and uniform boundedness principle, RadonNikodym theorem, Riesz Representation theorem, Lebesgue decomposition, Banach and Hilbert Spaces, BanachSteinhaus theorem, topological spaces, ArzelaAscoli and StoneWeierstrass theorems, compact sets and Tychonoff's theorem.

MATH858L  Mathematical Methods in Machine Learning Wojciech Czaja
Fall 2017

MATH630  Real Analysis I Radu Balan
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Lebesgue measure and the Lebesgue integral on R, differentiation of functions of bounded variation, absolute continuity and fundamental theorem of calculus, Lp spaces on R, RieszFischer theorem, bounded linear functionals on Lp, measure and outer measure, Fubini's theorem.

MATH634  Harmonic Analysis John J. Benedetto
Syllabus
1. Fourier analysis on Euclidean spaces including Fourier series
2. Distribution theory and applications
3. Fourier analysis on locally compact abelian groups
4. The uncertainty principle
5. Modern applications of Fourier analysis
6. Discrete Fourier series (DFT) and the Fast Fourier Transform (FFT)
5. Sampling theory and the relations between Fourier transforms,
Fourier series, and DFTs
6. Background for timefrequency analysis, wavelet theory on
Euclidean spaces and local fields, image processing, dimension reduction,
compressive sensing, Wiener's Generalized Harmonic Analysis, and frames
Spring 2017

MATH631  Real Analysis II Wojciech Czaja
Abstract measure and integration theory, metric spaces, Baire category theorem and uniform boundedness principle, RadonNikodym theorem, Riesz Representation theorem, Lebesgue decomposition, Banach and Hilbert Spaces, BanachSteinhaus theorem, topological spaces, ArzelaAscoli and StoneWeierstrass theorems, compact sets and Tychonoff's theorem.
Fall 2016

MATH630  Real Analysis I Wojciech Czaja
Lebesgue measure and the Lebesgue integral on R, differentiation of functions of bounded variation, absolute continuity and fundamental theorem of calculus, Lp spaces on R, RieszFischer theorem, bounded linear functionals on Lp, measure and outer measure, Fubini's theorem.

MATH858W  Wavelets, TimeFrequency Analysis, and Frames John J. Benedetto
Spring 2016
MATH858L  Mathematical Methods in Machine Learning Wojciech Czaja
Fall 2015
MATH634  Harmonic Analysis John J. Benedetto
Syllabus
1. The fundamental relation between Fourier analysis and number theory in topics such as the FFT, spectral synthesis, the padics, uniform distribution, Kronecker�s theorem, the HRT conjecture and the Riemann zeta function.
2. Carleson's theorem for Fourier series and recent related research.
3. Algebraic and geometric fundamentals of harmonic analysis, e.g., factorization and automorphisms of group algebras and the characterization of idempotent measures.
4. Beurling algebras, weighted norm inequalities, spectral analysis.
5. Statements and discussions of specific open problems and general unresolved issues: the uncertainty principle, MRI and nonuniform sampling, the Fuglede conjecture and the results of Tao, deterministic compressive sensing and the results of Bourgain, ambiguity functions and Wigner distributions, waveform design and the construction of sequences in terms of Weil�s solution of the Riemann hypothesis for finite fields, the characterization of the space of absolutely convergent Fourier transforms.
Spring 2015

MATH631  Real Analysis II Kasso Okoudjou
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Abstract measure and integration theory, metric spaces, Baire category theorem and uniform boundedness principle, RadonNikodym theorem, Riesz Representation theorem, Lebesgue decomposition, Banach and Hilbert Spaces, BanachSteinhaus theorem, topological spaces, ArzelaAscoli and StoneWeierstrass theorems, compact sets and Tychonoff's theorem.
Math858R  Selected Topics in Analysis: Sparse and Low Rank Representations: Analytical Methods and Industrial Applications Radu Balan
Course page
Topics inlude:
Approximation Principles: Nonparametric vs. parametric models, linear vs. nonlinear, deterministic vs. stochastic estimation.
Linear models: Stochastic approach: KarhunenLoev decmposition; PCA; ICA
Deterministic approach: case studies: spectral approximations; Fix
Strang conditions for shiftinvariant spaces; Blind source separation for sparse signals
Model Selection:
Classic principles: Akaike information criterion; Bayesian information
criterion; Minimum description length
Sparse Models
Nonlinear Models
Matrix Completion Problem
Phaseless reconstruction
Fall 2014
MATH630  Real Analysis I Kasso Okoudjou
Course Page
Lebesgue measure and the Lebesgue integral on R, differentiation of functions of bounded variation, absolute continuity and fundamental theorem of calculus, Lp spaces on R, RieszFischer theorem, bounded linear functionals on Lp, measure and outer measure, Fubini's theorem.
MATH858F  Selected Topics in Analysis: TimeFrequency and Wavelet Analysis  Theory and Applications John Benedetto
Topics include: Timefrequency (Gabor) analysis on R^d and the role of the Heisenberg group, Shorttime Fourier Transform, Wavelet theory on R^d, Timefrequency and wavelet uncertainty principles, the BailanLow phenomenon, Compressive sensing, Timefrequency and Wavelet frames, frame multiresolution analysis.
Spring 2014
Math858C  Selected Topics in Analysis: Geometric Multiresolution Representation Wojciech Czaja
A survey through multiresolution representations that capture geometric information. The "lets", including shearlets, curvelets, ridgelets, and contourlets, are discussed and analyzed.
Math858R  Selected Topics in Analysis: Sparse and Low Rank Representations: Analytical Methods and Industrial Applications Radu Balan
Course page
Topics inlude:
Approximation Principles: Nonparametric vs. parametric models, linear vs. nonlinear, deterministic vs. stochastic estimation.
Linear models: Stochastic approach: KarhunenLoev decmposition; PCA; ICA
Deterministic approach: case studies: spectral approximations; Fix
Strang conditions for shiftinvariant spaces; Blind source separation for sparse signals
Model Selection:
Classic principles: Akaike information criterion; Bayesian information
criterion; Minimum description length
Sparse Models
Nonlinear Models
Matrix Completion Problem
Phaseless reconstruction
Fall 2008



