Course detail
Applied Harmonic Analysis
FSI-9AHA Acad. year: 2024/2025 Summer semester
General theory of generating systems in Hilbert spaces: orthonormal bases (ONB), Riesz bases (RB), frames and reproducing kernels.
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.
Language of instruction
Czech
Supervisor
Department
Entry knowledge
Linear algebra, differential and integration calculus, linear functional analysis.
Rules for evaluation and completion of the course
Seminar presentations and/or oral examination.
Absence has to be made up by self-study and possibly via assigned homework.
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^2(R) (the space of analog signals) and L^2(Omega;A;P) (stochastic linear time series models).
Attention will be paid also to the problems of finding numerically stable sparse solutions in models with a large number of parameters.
Getting basic theoretical knowledge in modern harmonic analysis. Attaining practical skills which will allow the PhD students to use all these approaches effectively in computer-aided modeling and research of real systems.
The study programmes with the given course
Programme D-APM-K: Applied Mathematics, Doctoral, recommended course
Programme D-APM-P: Applied Mathematics, Doctoral, recommended course
Type of course unit
Lecture
20 hours, optionally
Syllabus
Facultative topics related to the students' doctoral study programe:
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. Reproducing kernel Hilbert spaces
6. Selected algorithms for the solution of inverse problems, handling numerical instability connected with overparametrization (overcomplete frames)
7. Some special spaces and their properties
8. Some special operators and their properties
9. Gabor frames
10. Wavelet frames
11. Multiresolution analysis
12. Reserve
Seminar: student presentations of special topics possibly closely connected with PhD thesis