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Course, academic year 2018/2019
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Approximation Theory 2 - NMNV568
Title in English: Teorie aproximace 2
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018
Semester: summer
E-Credits: 3
Hours per week, examination: summer s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Guarantor: doc. RNDr. Václav Kučera, Ph.D.
Class: M Mgr. NVM
M Mgr. NVM > Volitelné
Classification: Mathematics > Numerical Analysis
Annotation -
Last update: RNDr. Miloslav Vlasák, Ph.D. (10.05.2018)
The course is a follow-up of the Approximation Theory course and supplements selected important topics in approximation theory that do not fit in the winter course. The focus is especially on the basics of spline functions and wavelets.
Course completion requirements -
Last update: doc. RNDr. Václav Kučera, Ph.D. (12.05.2018)

The exam is oral. The examination requirements are given by the topics in the syllabus, in the extent to which they they were taught in course.

Literature - Czech
Last update: doc. RNDr. Václav Kučera, Ph.D. (13.05.2018)

NAJZAR K., Základy teorie splinů, Univerzita Karlova v Praze, Nakladatelství Karolinum, Praha, 2006.

MICULA G., MICULA S. Handbook of splines, Kluwer Academic Publishers, 1999.

FARIN G., Curves and surfaces for computer aided geometric design, Academic Press, 1990.

NAJZAR K., Základy teorie waveletů, Univerzita Karlova v Praze, Nakladatelství Karolinum, Praha, 2006.

DAUBECHIES I., Ten lectures on wavelets, CBMS-NSF Lecture Notes nr. 61, SIAM , 1992.

TREFETHEN N.L., Approximation Theory and Approximation Practice, SIAM, Philadelphia, PA, 2013.

RIVLIN T.J., An introduction to the approximation of functions, Blaisdell Publishing Co. Ginn and Co., 1969.

CHENEY E.W., Introduction to approximation theory, AMS Chelsea Publishing, Providence, RI, 1982.

Syllabus -
Last update: doc. RNDr. Václav Kučera, Ph.D. (12.05.2018)

Spline functions - polynomial splines, basic concepts and definitions. Interpolation and approximation properties. Qualitative properties - monotonicity and convexity preserving. Extremal properties of splines. Smoothing splines. Bézier curves, B-splines, rational B-splines.

Wavelets - Discrete Fourier transform, window Fourier transform, Haar basis, wavelet definition. Wavelet analysis, reconstruction and compression. Daubechies wavelets, 2D wavelets. Approximation properties.

Rational approximation: Interpolation, best approximation, continued fractions, Padé approximation.

Entry requirements -
Last update: doc. RNDr. Václav Kučera, Ph.D. (12.05.2018)

General knowledge of mathematical analysis. Basic knowledge of functional analysis. Passing the Approximation Theory course is welcome but not necessary.

 
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