Course detail

Applied Harmonic Analysis

FSI-9AHAAcad. year: 2016/2017Summer semesterNot applicable.. year of study2  credits

General theory of generating systems in separable Hilbert spaces: orthonormal bases (ONB), Riesz bases (RB) and frames.
The associated operators (for reconstruction, discretization, etc.). Properties and characterization theorems. Canonical duality. Useful constructions and algorithms based on the application of the theory of pseudoinverse operators. Special frames (Gabor and wavelet) and their applications.

Learning outcomes of the course unit

Getting basic theoretical knowledge in modern harmonic analysis. Attaining practical training which will allow the PhD students to use all these approaches effectively in computer-aided modeling and research of real systems.

Prerequisites

Linear algebra, differential and integration calculus, linear functional analysis.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

S.S. Chen, D.L. Donoho and M. Saunders: Atomic decomposition by Basis
O. Christensen: An Introduction to Frames and Riesz bases. Birkhäuser 2003
A. Teolis: Computational Signal processing with wavelets. Birkhäuser 1998
G.G. Walter: Wavelets and other orthogonal systems with Applications.
Ch. K. Chui: An Introduction to wavelets. Academic Press 1992
I. Daubechies: Ten Lectures on Wavelets. SIAM 1992
Y. Meyer: Wavelets and operators. Cambridge University Press 1992
H.G. Feichtinger and T. Strohmer: Gabor Analysis and Algorithms. Theory

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline.

Assesment methods and criteria linked to learning outcomes

Seminar presentations and oral examination.

Language of instruction

Czech, English

Work placements

Not applicable.

Aims

The PhD students will be made familiar with the latest achievements of the modern harmonic analysis and their applicability for the solution of practical problems of functional modeling in abstract spaces, in particular in l^2(J) (the space of discrete signals incl. images), L^(R) (the space of analog signals) and L^2(Omega;A;P) (stochastic linear time series models).
Attantion will be paid also to the problems of finding numerically stable sparse solutions in models with a large number of parameters.

Type of course unit

 

Lecture

20 hours, optionally

Teacher / Lecturer

Syllabus

1. Pseudoinverse operators in Hilbert spaces
2. Transition from orthonormal bases (ONB) to Riesz bases (RB) and frames
3. Discretization, reconstruction, correlation and frame operator
4. Characterizations of ONBs, RBs and frames. Duality principle
5. Selected algorithms for the solution of inverse problems, handling numerical instability connected with overparametrization (overcomplete frames)
6. Some special spaces and their properties
7. Some special operators and their properties
8. Gabor frames
9. Wavelet frames
10. Multiresolution analysis
11. Spare space
Seminar: student presentations of special topics possibly closely connected with PhD thesis