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

## Topics

1. Fourier series (6 weeks)
• Orthogonality
• Mean and pointwise convergence
• Fourier transforms
• Sampling theorem and uncertainty principle
• Fast Fourier transform
• Continuous and Discrete Filters
2. Orthogonal functions (1 week)
• Inner products
• Orthogonality and orthogonal subspaces
• Completeness, expansions, and mean convergence
• Examples
3. The Haar Wavelet (1 week)
• Haar functions
• Multiresolution analysis with Haar functions
4. Wavelets (4 weeks)
• Basic construction
• Multiresolution analysis
• Daubechies wavelets
• Discrete wavelet transform
• Continuous wavelet transform
• Nonorthogonal wavelets--Chui-Wang
• Multidimensional wavelets
5. Applications (2 weeks)
• Data compression
• Signal processing
• Fault (singularity) detection
• Image processing

## Texts & Articles

1. G. P. Tolstov, Fourier Series, Dover, New York, 1962.
2. C.-K. Chui, An introduction to wavelets, Academic Pess, New York, 1992.
3. I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992.
4. Y. Meyer,
5. Wavelets & Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1993.
6. R. S. Strichartz, ``How To Make Wavelets,'' Amer. Math. Monthly, 100 (1993), 539-556.
7. G. Strang, ``Wavelet transforms vs. Fourier transforms,'' Bull. Amer. Math. 28 (1993), 288-305.
8. 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\%].