Wavelets

 

Welcome to the course Wavelets, E2001.

Lectures are held on Wednesdays at 13.15-15.00 in Auditorium 7, beginning on September 5, 2001.

Lecturer: Prof. Gerd Grubb, whose office in the H.C.ěrsted Institute is E203, tel. 35320743, e-mail: grubb@math.ku.dk

Textbook: E. Hernandez - G. Weiss: A first course on wavelets, CRC Press 1996 *)**), possibly supplied with lecture notes if the need arises.

The course will be given in English as long as there are participants who do not speak Danish.

Notes on wavelets 2001, by Gerd Grubb, are distributed at the lectures. Filters and MRA

 

*) An alternative, the book by P. Wojtaszscyk: A Mathematical Introduction to Wavelets, Cambridge University Press 1997, has been inspected. It is OK, but slightly narrower in scope, so I'll stick to the original plan of using Hernandez-Weiss as the basic textbook. GG

**) The library copy is available in the secretaries' office E103 for inspection.

 

 

 

PLAN.

Past lectures:

Sept. 5: Pages 2-5 in Hernandez-Weiss. Repetition of facts about Fourier series and Fourier transforms. Extra example: The Str÷mberg wavelet.

Sept. 12: Pages 5-7 in H-W, the Shannon wavelet. Pages 44-46 in H-W, multiresolution analysis in the orthonormal case.

Sept. 19: Pages 46-53 (without Lemma 1.8 and its implications):The completeness property of an MRA. A condition for orthonormality. Start of the search for a wavelet associated with a scaling function. Homework No.1 (due Oct. 3).

 Sept. 26: Pages 49-50 on how to obtain an orthonormal scaling function from a Riesz scaling function. Pages 53-56 on the construction of a wavelet associated with a given scaling function.

Oct. 3: Lecture cancelled (postponed).

Oct. 10: Pages 57-58 on the wavelets associated with a scaling function. Pages 1-3 of notes, on filters.

Week 42: Vacation

Oct. 24: Pages 3-11 of notes, on the connection between filters and the projection operators in an MRA.

Oct. 31: Pages 9-15 of notes: The role of the filter operators, standard choices of the high-pass filter, the approximation theorem. Examples. Haar (with explicit calculation of filter mechanisms), Shannon, Daubechies 4 and 6.

Nov. 7: Pages 16-18 of notes: The cascade algorithm. Convergence of piecewise linear approximations. Pages 68-70, and 76-78 of H-W, about the construction of scaling functions from filters. Homework No. 2: Do two of the three exercises 3.1-3.3 on page 18 of the notes (due November 21)

Nov. 14: Lecture by Jens Gerlach Christensen on Franklin spline-wavelets (H-W Section 4.1)

Nov. 15 at 3.15 in aud. 9: Extra lecture. Continuation of construction of compactly supported wavelets in H-W: Page 79 on the support of \phi, and pages 82-85 on the construction of examples of g and extraction of a "square root".

Nov. 21: Remaining points in H-W, Section 2.3.

Nov. 28: Section 4.2 of H-W on higher order splines and spline-wavelets.

Dec. 5: In the first session, Nadia Larsen talked about connections of the filter formulation (Th. 2.1 in the notes) to Operator Algebra. Second session: Remaining points in Section 4.2. Constructions for bounded intervals (Section 4.5).

 Next:

Dec. 12: Section 4.5. Convergence of wavelet expansions (Ch. 5).