Approximation of functions 2

Course Journal

Last update: 02/06/2021

Lecture 1 - 02/03/21

Introduction. Why rational functions? Rational function representation. Refer to:

  • [TRE] Chapter 23.

Lecture 2 - 09/03/21

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.

Lecture 3 - 17/03/21

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

Lecture 4 - 24/03/21

Padé approximant. Definition, existence, uniqueness, Padé table. Refer to:

Lecture 5 - 31/03/21

Curves for CAGD. Introduction, parametric curves, de Casteljau algorithm, Bézier curve. Refer to:

Lecture 6 - 07/04/21

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).

Lecture 7 - 14/04/21

De Boor algorithm, B-spline segments and their derivatives, B-splines construction. Refer to:

  • [FAR] Section 8.1--8.3.

Lecture 8 - 21/04/21

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.

Lecture 9 - 28/04/21

Wavelets: mother function, dilatation, translation. Discrete wavelet transform. Continuous wavelet transform and its inverse. Multiresolution analysis, father function, dilatation equation.

  • [BUL] Section 3--5.

Lecture 10 - 05/05/21

Father function construction by Fourier transformation. Support of the father function. Interpretation of multiresolution. Mother function, orthogonal complement subspace, support.

  • [BUL] Section 6--8.

Lecture 11 - 19/05/21

Existence of wavelets (first part).

  • [BUL] Section 9 (without proofs), section 10 (until p. 21).

Lecture 12 - 26/05/21

Existence of wavelets (second part). Wavelet decomposition and reconstruction.

  • [BUL] Section 10 (from p. 22), section 11-12 (no proofs).

Lecture 13 - 02/06/21

Fast discrete wavelet transform (DWT). Truncated wavelet approximation. Image compressio.

  • [BUL] Sections 13, 14, 17, 18.

Exam:

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.

Materials and literature:

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):