#### 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

MATH 246 or MATH 341

#### 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, Jordan's theorem, and examples.

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

**Fourier series**

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.

Gibbs phenomenon.

**Laplace transform**

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.

**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 applicsations 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.

**DFT and FFT**

Definition and properties of the DFT.

Description of the FFT algorithm, and examples.

Applications with MATLAB.

**Signal processing**

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.

Multidimensional results.

Wavelet packets.

Applications with the MATLAB Wavelet Toolbox.

**Other transforms**

hankel, Hilbert, Mellin, and Radon transforms.