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) |

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:

- Applications to the storage of finger prints,
- The new JPEG200-standard.
- Some good introductory articles can be found here. webpage
- For applications in science, Engineering, Medicine, or Finance see e.g. The Illustrated Wavelet Transform Handbook by Paul S. Addison and references there.
- Application in computer graphics.
- A presentation Given by Imitiaz Hossain in my class 2003.

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:

- Why wavelets?
- Functions and approximation.
- The one-dimensional Haar wavelet.
- The two-dimensional Haar wavelet.
- The complex numbers
- Linear algebra:
- Vector spaces.
- Inner products.
- Linear independence and basis.
- Orthonormal basis and the connection with the wavelet transform.

- The Fourier transform
- 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!

There will be **3** graded assignments (about 6 problems
each) due

- Thursday, September 10.
- Thursday, October 8.
- Thursday, November 5.

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

- Thursday, September 17.
- Thursday, October 15.
- Thursday, November 12.

There will be **3** tests:

- Thursday, September 24.
- Thursday, October 22.
- Thursday, November 19.

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

- Home work: 30 each. Total: 90.
- Quizzes: 20 each. Total: 60.
- Tests: 100 each. Total: 300.
- Final: 150.
- Total points:
**600**.

**Grades:** A > 540, B > 480, C > 420, D ≥ 360, and F<360.