Course: 2020/2021

Hilbert spaces, wavelets and sampling theory

(15450)

1.- Knowledge of the main properties of Hilbert spaces.
2.- Using orthonormal bases, Riesz bases and frames in order to the stable recovery of elements in a Hilbert space.
3.- Knowledge of the main properties of Fourier transform.
4.- Introduction to Shannon's sampling theory.
5.- Reproducing kernel Hilbert spaces (RKHS)
5.- Introduction to wavelet theory.
6.- Construction of orthonormal bases of wavelets from a MRA.

Description of contents: programme

1.- Basic properties of Hilbert spaces.
2.- Orthogonal projection theorem.
3.- Orthonormal bases. Fourier series.
4.- Riesz bases and Frames.
5.- Fourier transform.
6.- Shannon's sampling theorem.
7.- Reproducing kernel Hilbert spaces.
8.- Multiresolution analyses.
9.- Orthonormla bases of wavelets.
10.-Continuous wavelet transform.

Learning activities and methodology

The docent methodology includes:
- Master classes where the knowledge that the students should acquire will be presented. The students will have written notes and the basic references in order to make easier their subsequent work
- Problem classes where problems proposed to the students will be solved.
- Partial controls.
- Final control.

Assessment System

- % end-of-term-examination 60
- % of continuous assessment (assigments, laboratory, practicals...) 40

Basic Bibliography

- A. García García. Bases en espacios de Hilbert: teoría de muestreo y wavelets. Sanz y Torres. 2014
- G. Bachman, L. Narici and E. Beckenstein. Fourier and Wavelet Analysis. Springer. 2000
- G. G. Walter. Wavelets and Other Orthogonal Systems with Applications. CRC Press. 1994
- M. W. Frazier. An Introduction to Wavelets through Linear Algebra. Springer. 1999
- O. Christensen. An Introduction to Frames and Riesz Bases. Birhauser. 2003
- P. Brémaud. Mathematical principles of Signal Processing. Springer. 2002
- R. Young. An Introduction to Nonharmonic Fourier Series. Academic Press. 2001
- S. Mallat. A Wavelet Tour of signal Processing. Academic Press. 2009