Approximation and complexity 1000-135APZ

A. Classical polynomial approximation

1. Problem formulation, a general characterization of optimal approximations
2. Approximation in Hilbert spaces
3. Algebraic and trigonometric polynomials, the Weierstrass theorem
4. Trigonometric approximation: Fourier and Fejer operators, Korovkin's theorem
5. Uniform approximation: Haar spaces, the Chebyshev theorem
6. Berntein's lethargy theorem
7. Theorems of Jackson and Bernstein

B. Information-based approximation

1. Information, error and cost of algorithms, problem complexity
2. Worst case setting: radius of information, optimality of linear algorithms
3. General splines and spline algorithms
4. Adaptive algorithms versus nonadaptive algorithms
5. Asymptotic setting
6. Randomization
7. Complexity of selected problems