Description

Fourier transform, Fourier series, discrete and fast Fourier transform (DFT and FFT). Laplace transform. Poisson summation, 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.

Prerequisites

1 course with a minimum grade of C- from (MATH246, MATH341).

Recommended: MATH240 or MATH 461 (strongly recommended)

Level of Rigor

Advanced

Sample Textbooks

Harmonic Analysis and Applications, 1st Edition by J.J. Benedetto

A First Course in Fourier Analysis, (2nd Edition) by D.W. Kammler

Applications

Chemistry, engineering, physics and astronomy

If you like this course, you might also consider the following courses

MATH 416, STAT 426, MATH 420, ENEE 322

Additional Notes

Topics

Fourier Transform

Algebraic properties of the Fourier transform: convolution, modulation, and translation.

Analytic properties of the Fourier transform: Riemann-Lebesgue Lemma, transforms of derivatives, and derivatives of transforms.

Inversion theory: Approximate identities, L1 inversion, and examples.

The L2 theory: Parseval's formula, Plancherel's theorem, and examples.

Fourier series

Pointwise convergence theory: Dirichlet's theorem and examples; absolute convergence; Gibbs phenomenon.

Differentiation and integration of Fourier series.

The L1 and L2 theories.

Review of complex variables.

Differential equations

Applications of Fourier transforms, Fourier series, and Laplace transforms to ODE's and PDE's. These include recent applications in signal processing, classical applications in mathematical physics, initial and boundary value problems, Bessel functions, etc.

Distribution theory

Motivation, definitions, elementary results, and examples.

Fourier transforms of distributions.

Convolution equations.

Linear translation invariant systems.

Operational calculus.

Applications with MATLAB.

Signal processing

Poisson summation and applications.

The classical sampling theorem.

Uncertainty principle and entropy inequalities.

Sampling of Band-limited functions. 

Shannon sampling formula.

Classical sampling theorem.

Multiresolution and analysis wavelet orthonormal bases.

Other transforms:

Hankel, Hilbert, Mellin, and Radon transforms.