NWC Courses

Fall 2019

• MATH464 - Transform Methods for Scientists and Engineers
  Radu Balan

  Course Page
  
  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 Low-Rank 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 time-frequency 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, Riesz-Fischer 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

  Course Page
  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, Radon-Nikodym theorem, Riesz Representation theorem, Lebesgue decomposition, Banach and Hilbert Spaces, Banach-Steinhaus theorem, topological spaces, Arzela-Ascoli and Stone-Weierstrass theorems, compact sets and Tychonoff's theorem.

Fall 2018

• MATH858W - Frames, Sampling, and Wavelets
  John J. Benedetto

  Syllabus
  
  1. Time-frequency (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. Time-frequency (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, quasi-crystals and non-uniform Poisson summation, balayage and nonuniform sampling, Sigma-Delta quantization and non-linear 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 Balian-Low phenomenon and Bourgain's theorem.
  

• MATH632 - Functional Analysis
  Radu Balan

  Course Page
  Abstract measure and integration theory, metric spaces, Baire category theorem and uniform boundedness principle, Radon-Nikodym theorem, Riesz Representation theorem, Lebesgue decomposition, Banach and Hilbert Spaces, Banach-Steinhaus theorem, topological spaces, Arzela-Ascoli and Stone-Weierstrass theorems, compact sets and Tychonoff's theorem.

Spring 2018

• MATH631 - Real Analysis II
  Radu Balan

  Course Page
  Abstract measure and integration theory, metric spaces, Baire category theorem and uniform boundedness principle, Radon-Nikodym theorem, Riesz Representation theorem, Lebesgue decomposition, Banach and Hilbert Spaces, Banach-Steinhaus theorem, topological spaces, Arzela-Ascoli and Stone-Weierstrass theorems, compact sets and Tychonoff's theorem.

• MATH858L - Mathematical Methods in Machine Learning
  Wojciech Czaja

Fall 2017

• MATH630 - Real Analysis I
  Radu Balan

  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, Riesz-Fischer 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 time-frequency 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, Radon-Nikodym theorem, Riesz Representation theorem, Lebesgue decomposition, Banach and Hilbert Spaces, Banach-Steinhaus theorem, topological spaces, Arzela-Ascoli and Stone-Weierstrass 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, Riesz-Fischer theorem, bounded linear functionals on Lp, measure and outer measure, Fubini's theorem.

• MATH858W - Wavelets, Time-Frequency 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 p-adics, 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 non-uniform 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

  Course Page
  Abstract measure and integration theory, metric spaces, Baire category theorem and uniform boundedness principle, Radon-Nikodym theorem, Riesz Representation theorem, Lebesgue decomposition, Banach and Hilbert Spaces, Banach-Steinhaus theorem, topological spaces, Arzela-Ascoli and Stone-Weierstrass 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: Non-parametric vs. parametric models, linear vs. nonlinear, deterministic vs. stochastic estimation.
  Linear models: Stochastic approach: Karhunen-Loev decmposition; PCA; ICA Deterministic approach: case studies: spectral approximations; Fix- Strang conditions for shift-invariant 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, Riesz-Fischer theorem, bounded linear functionals on Lp, measure and outer measure, Fubini's theorem.

• MATH858F - Selected Topics in Analysis: Time-Frequency and Wavelet Analysis - Theory and Applications
  John Benedetto

  Topics include: Time-frequency (Gabor) analysis on R^d and the role of the Heisenberg group, Short-time Fourier Transform, Wavelet theory on R^d, Time-frequency and wavelet uncertainty principles, the Bailan-Low phenomenon, Compressive sensing, Time-frequency 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: Non-parametric vs. parametric models, linear vs. nonlinear, deterministic vs. stochastic estimation.
  Linear models: Stochastic approach: Karhunen-Loev decmposition; PCA; ICA Deterministic approach: case studies: spectral approximations; Fix- Strang conditions for shift-invariant 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

• Lecture video from MAIT 679W/MATH 648W: Wavelet Theory and Waveform Design