# Math 2025

T-TH 9:10-10:30, Lockett 112

 INSTRUCTOR Gestur Olafsson Office 322 Lockett Office Hours T-TH 1:30 --2:30 PM and by request Phone 578-1608 and 225-337-2206 (cell) e-mail olafsson@math.lsu.edu Internet http://math.lsu.edu/~olafsson Text: Wavelets Made Easy by Yves Nievergelt

The theory of wavelets is a relatively recent mathematical theory. It is the basic theory behind several modern applications in storage of electronic information, data compression, image reconstruction and electronic transmission of information. Here are some other interesting links:

The basic ideas can be formulated using the language of linear algebra: Vector spaces, subspaces, linear maps, inner product, orthogonal projections, and basis. Related related concept in analysis are: Vector spaces of functions, approximation of functions using basic functions (in our case wavelets), dilation and translation, change of basis. We will follow the book Wavelets Made Easy but also use the lecture notes on linear algebra.

The outline of the course is:

1. Why wavelets?
2. Functions and approximation.
3. The one-dimensional Haar wavelet.
4. The two-dimensional Haar wavelet.
5. The complex numbers
6. Linear algebra:
• Vector spaces.
• Inner products.
• Linear independence and basis.
• Orthonormal basis and the connection with the wavelet transform.
7. The Fourier transform
8. If there is still time, then we will discuss some other wavelets.

The wavelet transform is only one example of integral transforms. The Fourier Transform is much older, almost 200 years old, and still widely used. It has become an indispensable tool in mathematics and applied sciences. We will discuss some aspects of the Fourier Transform starting with the Fast Fourier transform. Before doing that, we will need to introduce the field of complex numbers, and the complex exponential function.

If there is time we will look at Daubechies Wavelets from Chapter 3, and How to Make Wavelets in general, following R.S. Strichartz's paper in The American Mathematical Monthly, 100, No. 6 (June - July, 1993), pp. 539-556.

To do well in a math class, it is essential to get sufficient practice working problems. Accordingly, we will have both graded and ungraded homework assignments. You should therefore work on the exercises in the book, even if they will not be collected or graded. But I will discuss some of them in class. Those problems are similar to those on the tests, so work on it!

1. Thursday, September 10.
2. Thursday, October 8.
3. Thursday, November 5.

There will be 3 short quizzes in class. Each 2 to 3 problems.

1. Thursday, September 17.
2. Thursday, October 15.
3. Thursday, November 12.

There will be 3 tests:

1. Thursday, September 24.
2. Thursday, October 22.
3. Thursday, November 19.

The final takes place Tuesday, Dec. 8, 7:30-9:30, in the same room as class.

Absence on a Test, Quizz or Final makes automatically 0 points. Only serious and verifiable excuses will be respected.