Approximation Theory 1000-135TAP

The scope of approximation theory: existence and uniqueness of the best approximations, convergence and rate of convergence of sequences of approximations with given properties, estimates for error of approximation, basic concepts in approximation theory, characterization of the best approximations using dual spaces. Examples: strongly, uniformly and strictly convex spaces, approximation in Hilbert spaces.

Dense subsets in function spaces: algebraic and trigonometric polynomials, splines, etc, Weierstrass theorems.

Trigonometric approximation: Fourier and Fejer operators and their properties, Lozinski - Harsziladze and Korowkin theorems.

Approximation in C(K) spaces: Haar subspaces, Kolmogorov criterion, alteration theorem, Remez algorithms. Trigonometric interpolation.

Jackson and Bernstein theorems for trigonometric and algebraic polynomials. Regularity of functions vs. distance from subspaces of polynomials of specified degree.

Lethargy theorem.

Schauder bases and unconditional bases in Banach spaces.

Nonlinear approximation: greedy approximation.

Type of course

elective courses

Prerequisites

Functional Analysis I

Prerequisites (description)

Basic functional analysis knowledge. Notions of normed, Banach and Hilbert spaces, spaces of continuous and smooth functions, Lebesgue spaces (L_p), continuous linear functionals, dual space, continuous linear operators. Fundamental theorems of Banach space theory. Orthogonal bases in Hilbert spaces.

Learning outcomes

Knowledge and skills:
The student
1. knows the object of approximation theory and basic types of problems posed by it,
2. knows existence and uniqueness theorems for the best approximations in various Banach spaces
3. knows examples of linearly dense sets in various Banach spaces
4. knows main theorems about approximation in spaces of continuous functions
5. is capable of using notions and theorems concerning trigonometric and polynomial approximation
6. knows Bernstein's lethargy theorem and its practical and theoretical implications.

Social skills:
The student
1. Understand the significance of approximation of functions in information technology, engineering and natural sciences.

Assessment criteria

Final grade is based on participation in classes, home assignments, written egzams (mid-term and final) and/or oral exam.

Bibliography

E. W. Cheney, Introduction to Approximation Theory, AMS 2000.

O. Christensen, K. L. Christensen, Approximation Theory, Birkhauser 2004.

C. Heil, A Basis Theory Primer, Birkhauser 2011.

V. Temlyakov, Greedy Approximation, Cambridge 2011

Additional information

Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system:

Description of 1000-135TAP in USOSweb