Math 641-600 Midterm Review (Fall 2011)

Math 641-600 Midterm Review

The midterm will be given on Wednesday, Oct. 19 and will consist of an in-class part and a take-home part. It will cover sections 1.1-1.4, 2.1, 2.2.1-2.2.4. It will also cover the material done in class on the Lebesgue integral, point-wise convergence of Fourier series, and the material covered from my notes on the Discrete Fourier Transform. The in-class part of the midterm will consist of the following: statements of theorems and definitions; short problems or propositions similar to homework problems or examples done in class; and either a critical part or sketch of a proof for one of the major theorems proved. The take-home test will have longer computations, proofs, or problems.

Linear algebra

Inner product spaces & normed linear spaces

Subspaces, orthogonal complements
Orthogonal sets of vectors, the Gram-Schmidt procedure
Least squares, minimization problems, projections, normal equations

Self-adjoint matrices & their properties

Estimation of eigenvalues

Rayleigh-Ritz maximum principle
The Courant-Fischer Minimax Theorem and applications

The Fredholm Alternative

Function spaces

Complete normed spaces & complete inner product spaces

Convergent sequence, Cauchy sequence, complete spaces - Hilbert spaces and Banach spaces
Special (complete) spaces - l^p, L^p (1 ≤ p ≤ ∞), C[a,b], C^k[a,b], Sobolev space Hⁿ[a,b], Sobolev-type inequalities
Dense sets in a Banach or Hilbert space; density of linear splines in C[a,b]
Separable vs. non-separable spaces

Lebesgue integral

Lebesgue measure, measurable functions, Lebesgue sums and Lebesgue integral
Density of continuous functions in L^p[a,b], 1 ≤ p < ∞
Dominated Convergence Theorems

Hilbert spaces & complete orthogonal sets

Minimization problems, least squares, normal equations
Complete sets of orthogonal/orthonormal functions, Parseval's identity, other conditions equivalent to completeness of a set

Weierstrass Approximation Theorem

Bernstein polynomials, modulus of continuity
Density of polynomials in L^p, 1 ≤ p < ∞
Completeness of orthogonal polynomials in L² and completeness of trigonometric functions/Fourier series in L²

Approximation tools

Orthogonal polynomials, Fourier series, and discrete Fourier transform/FFT

Updated 10/17/2011 (fjn).