Lecture outline for Approximation Theory

Approximation Theory

Lectures for Fall 2005

Introduction to Concepts of Approximation Theory

Week 1
Lecture 1 (8/18): Main Issues of Approximation Theory:

Choice of basic building components used for approximation:
- conditioned on problem being studied, e.g the physics in solving PDE,
- analysis and synthesis capabilities.
- efficiency of representation for a given function class
Sparse and computationally efficient representations
Existence/Uniqueness of Best Approximation
Measurement of the Quality of Approximation - choice of metric space
Near-best approximations
Linear vrs nonlinear approximation.

Defns and Preliminaries: norms, normed linear spaces, completeness; Simple examples of Banach spaces: C[a,b], C(T), L². Overview description of algebraic and trigonometric polynomials, splines, wavelets.

Space of Continuous Functions

Week 2
Lecture 2 (8/23): Generalized polynomials and the error of approximation of degree n (approximation numbers). Properties of continuous functions, uniform continuity; completeness of C(A); the modulus of continuity and Lipschitz spaces.
Lecture 3 (8/25): Convolution, approximate identities and summability kernels on the circle; Example of Gaussian kernel; Weierstrass' theorem on an interval.

Hilbert Spaces and General Fourier Series

Week 3
Lecture 4 (8/30): Elementary separable Hilbert space: inner product, Cauchy-Schwartz inequality, & triangle inequality; orthogonality, generalized Pythagorean theorem, the Fourier projection and best approximation from a finite dimensional linear subspace; Existence and Uniqueness of Best Approximation. Bessel's inequality, Parseval and Plancherel equations. Uniqueness theorem.

Classical Fourier Series

Lecture 5 (9/1): Gram-Schmidt orthonormalization and existence of complete orthonormal systems. Norm convergence of N-th Fourier partial sums in Hilbert space. Representation of separable Hilbert spaces in terms of l²(N). Classical examples: l²(Z), complex Fourier series on the circle; classical Fourier sine and cosine series on an interval.

Jackson's Theorem for Trigonometric Polynomials

Week 4
Lecture 6 (9/6): Relationship of trigonometric and algebraic polynomials. Trigonometric identities; N-th Fourier partial sum as convolution with the Dirichlet kernel; the Jackson kernels, asymptotic estimates of their moments, and representation as trig polynomials.
Lecture 7 (9/8): Jackson kernels as summability kernels; properties of modulus of continuity; Jackson's theorem for trigonometric polynomials; Rates of convergence for Lipschitz spaces; adjustments for the interval [a,b] and a non-optimal Jackson theorem for algebraic polynomials.

Week 5
Lecture 8 (9/13): Extensions of Jackson's theorem to L^p norms, rates of approximation for special Besov spaces (Lipschitz spaces based on L^p). Extension to higher smoothness and Lip_a(T), 1\leq a \leq 2. Additional properties of higher differences and moduli of smoothness (C and L^p).

Existence/Uniqueness and Characterization of Best Uniform Approximants

Lecture 9 (9/15): Existence of best (generalized polynomial) approximants in normed linear spaces; Compactness of closed and bounded sets in finite dimensional normed linear spaces; equivalence of norms in finite dimensional spaces. Strict convexity and a sufficient condition for uniqueness.

Characterization of Best Uniform Approximants - Case of Algebraic Polynomials

Week 6
Lecture 10 (9/20): Introduction to the Characterization of Best Approximants in the uniform norm, Chebyshev alterations. Review for Test #1.

Test 1 (9/22).

Week 7
Lecture 11 (9/27): Completion of the proof of Chebyshev's characterization of best approximation by algebraic polynomials in C[a,b]. Uniqueness of best approximation by algebraic polynomials in C[a,b].

Postponed (9/29): Class to be made up. NGA PI meeting.

Characterization of Best Uniform Approximants - Chebyshev Systems

Week 8
Lecture 12 (10/4): Review of Chebyshev Characterization for best uniform approximants in order to motivate the concept of Chebyshev systems of generalized polynomials. Definition, elementary properties, and examples of Chebyshev systems. Construction of generalized polynomials with prescribed 'simple zeros'.

Lecture 13 (10/6): Further properties of Chebyshev systems. Kolmogorov Characterization of best uniform approximation by generalized polynomials.

Week 9
Lecture 14 (10/11): Review and completion of the proof of Kolmogorov's Characterization Theorem. Construction of generalized polynomials from Chebyshev systems with particular properties.

Fall Break (10/13).

Week 10
Lecture 15 (10/18): Special case of Krein's theorem - generalized polynomials with specified 'simple' and 'double' zeros. Construction and properties of classical Chebyshev polynomials on [-1,1].

The Bernstein and Markov's Inequalities

Lecture 16 (10/20): Lagrange interpolation by trigonometric polynomials. Bernstein's inequality for trigonometric polynomials.

Week 11
Lecture 17 (10/25): Corollaries and extensions of Bernstein's inequality. Tightness of inequalities (i.e. best constants). Preliminary inequality for algebraic polynomials.

Lecture 18 (10/27): Markov's inequality for algebraic polynomials. Schur's theorem.

Smoothness Classes - the Besov spaces

Week 12
Lecture 19 (11/1): Definition of Besov spaces B_a^q(L^p) and elementary properties: Besov spaces as sequence spaces, embeddings (part I), Marchaud's inequality.

Lecture 20 (11/3): Hardy inequalities for averages. Embeddings of Besov spaces (Part II).

Inverse Theorems of Trigonometric Approximation

Week 13
Lecture 21 (11/8): Bernstein's inverse theorem. Equivalence of conditions on approximation decay rates in terms of smoothness of the target function.

Lecture 22 (11/10): TBD

Week 14
Lecture 23 (11/15): Review for Test #2.

Test 2 (11/17):