Approximation Theory
Lectures for Fall 2005
Introduction to Concepts of Approximation Theory
Space of Continuous Functions
Hilbert Spaces and General Fourier Series
Classical Fourier Series
Jackson's Theorem for Trigonometric Polynomials
Existence/Uniqueness and Characterization of Best Uniform Approximants
Characterization of Best Uniform Approximants - Case of Algebraic Polynomials
Characterization of Best Uniform Approximants - Chebyshev Systems
The Bernstein and Markov's Inequalities
Smoothness Classes - the Besov spaces
Inverse Theorems of Trigonometric Approximation
Lecture 1 (8/18): Main Issues of Approximation
Theory:
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.
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.
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.
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.
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.
Lecture 8 (9/13):
Extensions of Jackson's theorem to Lp norms, rates of approximation
for special Besov spaces (Lipschitz spaces based on Lp). Extension
to higher smoothness and Lipa(T), 1\leq a \leq 2. Additional properties
of higher differences and moduli of smoothness (C and Lp).
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.
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).
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.
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.
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).
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].
Lecture 16 (10/20): Lagrange interpolation by
trigonometric polynomials. Bernstein's inequality for
trigonometric polynomials.
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.
Lecture 19 (11/1): Definition of Besov spaces
Baq(Lp) 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).
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
Lecture 23 (11/15): Review for Test #2.
Test 2 (11/17):