| DESCRIPTION |
Option 1: Fourier
transform, Fourier series,
discrete
and fast Fourier transform (DFT and FFT). Distributions and
operational
calculus. Poisson summation, sampling, and appications.
Wavelet
teory and MATLAB applications.
Option 2: Fourier series, Fourier and Laplace
transforms.
Evaluation of the complex inversion integral by the theory of
residues.
Applications to partial differential equations, with solutions computed
by transforms and separation of variables. Bessel functions, and
Hankel and Mellin transforms. |
| TOPICS |
Option 1: Fourier transform and Fourier series
components.
Distribution theory component. DFT and FFT component. Wavelet
theory
and MATLAB component.
Option 2: Fourier transform and Fourier series
components.
Distribution theory component. Laplace transform component.
Other transforms component.
These components are now described more fully.
COMPONENTS:
Fourier Transform Component
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 component
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 component
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 component
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 component
Motivation, definitions, elementary results,
and examples.
Fourier transforms of distributions.
Convolution equations.
Linear translation invariant systems.
Operational calculus.
DFT and FFT component
Definition and properties of the DFT.
Description of the FFT algorithm, and
examples.
Applications with MATLAB.
Signal processing component
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 component
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 component
hankel, Hilbert, Mellin, and Radon transforms.
|
| TEXT |
Text(s)
typically used in this course.
Additional Information on texts:
Option 1:
Harmonic analysis and applications - by J.J.
Benedetto.
The Fourier transform and its applications,
by R.N.
Bracewell
MATLAB Wavelet Toolbox.
Fast Fourier transforms, by J.S. Walker
Option 2:
Harmonic analysis and applications by J.J. Benedetto (for
Fourier analysis).
The Fourier transform and its applications, by R.N.
Bracewell
Fourier series and boundary value problems, by J.Brown,
and R.V. Churchill
Operational mathematicsby R.V. Churchill (for Laplace
transforms)
NOTE: Churchill's book is excellent for the Laplace
transform
and classical applications of the Laplace transform. It is
inappropriate
for the Fourier transform.
|
