Introduction. Why rational functions? Rational function representation. Refer to:
[TRE] Chapter 23.
Rational best approximation, existence and characterization (equioscillation). Introduction to rational interpolation, existence, spurious poles. Refer to:
[TRE] Chapter 24, pp. 259-263 (without proof of Theorem 24.1) .
[TRE] Chapter 26, pp. 287-289.
Rational interpolation, linearized problem, solution uniqueness, unattainable points. A method for rational interpolation on the unit circle. Refer to:
[TRE] Chapter 26, pp. 290-293.
Uniqueness theorems: see lecture video
Padé approximant. Definition, existence, uniqueness, Padé table. Refer to:
Brezinski, Claude, and J. Van Iseghem. "A taste of Padé approximation." Acta numerica 4 (1995): 53-103. pp. 53-57.
[TRE] Chapter 27, Theorem 27.1
Curves for CAGD. Introduction, parametric curves, de Casteljau algorithm, Bézier curve. Refer to:
[FAR] Section 4.1-4.3.
A History of Curves and Surfaces in CAGD. (Extra reading if you are interested).
Bézier curve and Bernstein polynomials. General idea of blossom. Bézier curve properties by Bernstein polynomial representation. Derivative of a Bézier curve Refer to:
[FAR] Section 5.1--5.3, until p. 65.
[FAR] Section 4.4 (blossoms).
De Boor algorithm, B-spline segments and their derivatives, B-splines construction. Refer to:
[FAR] Section 8.1--8.3.
B-splines: locality, endpoint interpolation, and smoothness. Idea of rational Bézier curves. Refer to:
[FAR] Section 8.8 and 13.1.
Wavelets: motivation. Basics of Fourier transform.
[BUL] Section 1 and 2.
Wavelets: mother function, dilatation, translation. Discrete wavelet transform. Continuous wavelet transform and its inverse. Multiresolution analysis, father function, dilatation equation.
[BUL] Section 3--5.
Father function construction by Fourier transformation. Support of the father function. Interpretation of multiresolution. Mother function, orthogonal complement subspace, support.
[BUL] Section 6--8.
Existence of wavelets (first part).
[BUL] Section 9 (without proofs), section 10 (until p. 21).
Existence of wavelets (second part). Wavelet decomposition and reconstruction.
[BUL] Section 10 (from p. 22), section 11-12 (no proofs).
Fast discrete wavelet transform (DWT). Truncated wavelet approximation. Image compressio.
[BUL] Sections 13, 14, 17, 18.
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:
[TRE] TREFETHEN N.L., Approximation Theory and Approximation Practice, SIAM, Philadelphia, PA, 2013.
[FAR] FARIN G., Curves and surfaces for computer aided geometric design, Academic Press, 5th ed, 2001.
[BUL] BULTHEEL A., "Learning to swim in a sea of wavelets." Bulletin of the Belgian Mathematical society-simon stevin 2.1 (1995): 1-45.
NAJZAR K., Základy teorie splinů, Univerzita Karlova v Praze, Nakladatelství Karolinum, Praha, 2006.
MICULA G., MICULA S. Handbook of splines, Kluwer Academic Publishers, 1999.
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.
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.
Material discussed in class (in order of appearance):
Brezinski, Claude, and J. Van Iseghem. "A taste of Padé approximation." Acta numerica 4 (1995): 53-103