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.
MATH 246 or MATH 341
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, Jordan's theorem, and examples.
The L2 theory: Parseval's formula, Plancherel's theorem, and examples.
Representation theory: Dirichlet's theorem and examples.
Differentiation and integration of Fourier series.
The L1 and L2 theories.
Absolutely convergent Fourier series and Wiener's inversion theorem.
Review of complex variables.
Algebraic properties of the Laplace transform.
Analytic properties of the Laplace transform: regions of convergence, transforms of derivatives, and derivatives of transforms.
Representation and inversion theory of the Laplace transform.
Evaluation of the complex inversion formula by residues.
Applications of Fourier transforms, Fourier series, and Laplace transforms to ODE's and PDE's. These include recent applications in signal processing, classical applicsations in mathematical physics, initial and boundary value problems, Bessel functions, etc.
Motivation, definitions, elementary results, and examples.
Fourier transforms of distributions.
Linear translation invariant systems.
DFT and FFT
Definition and properties of the DFT.
Description of the FFT algorithm, and examples.
Applications with MATLAB.
Poisson summation and applications.
The classical sampling theorem.
Uncertainty principle and entropy inequalities.
Temporal and spectral widths.
Power spectrum: definitions, estimation, calculations, and examples.
Maximum entropy and linear prediction.
Wavelet theory and MATLAB
Shannon wavelets and the classical sampling theorem.
Multiresolution and analysis wavelet orthonormal bases.
Quadrature mirror filters and perfect reconstruction filter banks.
Applications with the MATLAB Wavelet Toolbox.
hankel, Hilbert, Mellin, and Radon transforms.