Math 641-600 Midterm Review
The midterm will be given on Thursday, Oct. 21, from 7 to 9 pm. It
will cover sections 1.1-1.4, 2.1, 2.2.1-2.2.4, 2.2.7. It will also
cover the material done in class on the Lebesgue integral.
The test will consist of the following: statements and/or proofs or
sketches of proofs of theorems; definitions; short problems or propositions
similar
to homework
problems or examples done in class.
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 - lp, Lp (1 ≤
p ≤ ∞), C[a,b], Ck[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]
- Lebesgue integral
- Lebesgue measure, measurable functions, Lebesgue sums and
Lebesgue integral
- Density of continuous functions in Lp[a,b], 1 ≤ p <
∞
- Monotone and Dominated Convergence Theorems
- Hilbert spaces & complete orthogonal sets
- Minimization problems, least squares - discrete and continuous -,
variational/finite element methods, 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 Lp, 1 ≤ p < &infin
- 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
- Splines and finite elements
- B-spline, Nm; knot sequence, t = {a =
t0 < ... < tn = b}; S(k,r,t) and
S1/n(k,r)
- Hermite cubic splines, φj, ψj;
finite element methods and curve fitting
Updated 10/18/2010 (fjn).