# Fourier Series and Wavelets--Math 414-500--Spring 1997

## Topics

- Fourier series (6 weeks)
- Orthogonality
- Mean and pointwise convergence
- Fourier transforms
- Sampling theorem and uncertainty principle
- Fast Fourier transform
- Continuous and Discrete Filters

- Orthogonal functions (1 week)
- Inner products
- Orthogonality and orthogonal subspaces
- Completeness, expansions, and mean convergence
- Examples

- The Haar Wavelet (1 week)
- Haar functions
- Multiresolution analysis with Haar functions

- Wavelets (4 weeks)
- Basic construction
- Multiresolution analysis
- Daubechies wavelets
- Discrete wavelet transform
- Continuous wavelet transform
- Nonorthogonal wavelets--Chui-Wang
- Multidimensional wavelets

- Applications (2 weeks)
- Data compression
- Signal processing
- Fault (singularity) detection
- Image processing

## Texts & Articles

- G. P. Tolstov, Fourier Series, Dover, New York, 1962.
- C.-K. Chui, An introduction to wavelets, Academic Pess, New
York, 1992.
- I. Daubechies, Ten Lectures on Wavelets, Society for
Industrial and Applied Mathematics, Philadelphia, PA, 1992.
- Y. Meyer,
- Wavelets & Applications, Society for Industrial
and Applied Mathematics, Philadelphia, PA, 1993.
- R. S. Strichartz, ``How To Make Wavelets,'' Amer. Math. Monthly,
**100** (1993), 539-556.
- G. Strang, ``Wavelet transforms vs. Fourier transforms,'' Bull.
Amer. Math.
**28** (1993), 288-305.
- S. Mallat, ``A theory for multi-resolution approximation: the wavelet
approximation,'' IEEE Trans. PAMI 11 (1989), 674-693.

## Prerequisites:

The course requires a solid grounding in linear algebra, especially inner
product spaces. A decent knowledge of sequences and series is also
important. Computer literacy will be assumed. Familiarity with
Maple or Matlab will be helpful.
## Method of Evaluation:

Students will be evaluated on the basis of a project [20%], homework
[15%], two in-class tests [20% each], and a final [25\%].