Applied Harmonic Analysis
FSI-9AHAAcad. year: 2020/2021
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.
Learning outcomes of the course unit
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.
Linear algebra, differential and integration calculus, linear functional analysis.
Recommended optional programme components
Recommended or required reading
V. Veselý a P. Rajmic. Funkcionálnı́ analýza s aplikacemi ve zpracovánı́ signálů, Odborná učebnice (4.vyd.). Vysoké učenı́ technické v Brně, Brno (CZ), 2019. ISBN 978-80-214-5186-5. (CS)
S.S. Chen, D.L. Donoho and M. Saunders: Atomic Decomposition by Basis Pursuit, SIAM J. Sci. Comput. 20 (1998), no. 1, 33–61, reprinted in SIAM Review, 43 (2001), no. 1, pp. 129–159. (EN)
Ch.Heil: A Basis Theory Primer, expanded edition, Birkhäuser, New York, 2011 (EN)
G.G. Walter: Wavelets and other orthogonal systems with Applications, CRC Press, Boca Raton, Florida, 1994. (EN)
O. Christensen: An Introduction to Frames and Riesz bases. Birkhäuser 2003 (EN)
Ch. K. Chui: An Introduction to wavelets, Wavelet Analysis and Its Applications, vol. 1, Academic Press, Inc., San Diego, CA, 1992. (EN)
A. Teolis: Computational Signal processing with wavelets. Birkhäuser 1998 (EN)
I. Daubechies: Ten Lectures on Wavelets, Ingrid Daubechies, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, SIAM, Philadelphia, Pennsylvania, 1992. (EN)
Y. Meyer: Wavelets and operators, Cambridge Studies in Advanced Mathematics, vol. 37, Cambridge University Press, Cambridge, 1992. (EN)
H. G. Feichtinger (ed.) and T. Strohmer (ed.), Gabor analysis and algorithms. Theory and applications, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston-Basel-Berlin, 1998 (EN)
Planned learning activities and teaching methods
The course is taught through lectures and/or seminars targeted at selected topics of the discipline.
Assesment methods and criteria linked to learning outcomes
Seminar presentations and/or oral examination.
Language of instruction
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.
Specification of controlled education, way of implementation and compensation for absences
Absence has to be made up by self-study and possibly via assigned homework.
Type of course unit
20 hours, optionally
Teacher / Lecturer
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
Seminar: student presentations of special topics possibly closely connected with PhD thesis