Math 641-600 Midterm Review — Fall 2021
The midterm will consist of an in-class part, which will be given on
Wednesday, 10/6/2021, and a take-home part. It will cover sections
2.1, 2.2.1-2.2.4. It will also cover the material done in class and
from the class notes
on my web page, up to and including the notes on Fourier series. I
will also have extra office hours next week. My office hours will be:
Monday, 1-2 & 2:30-3:30; Tuesday, 10-11 & 1-2; Wednesday,
9:30-10.
The in-class part of the midterm will consist of the following:
statements of theorems and definitions; short problems and
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. It will be due at 4 pm on Monday, October 11, 2021.
- Inner product spaces
- Definitions of inner product spaces and normed spaces
- Schwarz's inequality, triangle inequality
- Orthogonal projections, minimization problems, least squares,
normal equations
- Gram-Schmidt process
- Banach
spaces and Hilbert spaces
- Convergent sequences, Cauchy sequences, complete spaces - Hilbert
spaces and Banach spaces.
- Special (complete) spaces — $\ell^p, L^p\ (1 \le p \le
\infty),\ C[a,b], C^k[a,b]$, and Sobolev space $ H^1[a,b]$. Be able
to prove completeness for the spaces given in the notes or in
homework assignments.
- Lebesgue Integration
- Definitions: Lebesgue measure, Lebesgue integral, simple function, sets of
measure 0
- Be able to define the Lebesgue integral using simple functions.
- Be able to state and use these theorems: Monotone convergence
theorem and dominated convergence theorem
-
Orthonormal sets and expansions
- Complete sets of orthogonal/orthonormal functions, Parseval's
identity, other conditions equivalent to completeness of a set
- Dense sets and completeness
- Completeness of polynomials in L2, orthogonal
polynomials; be able to establish completeness for specific sets of
orthogonal polynomials.
-
Approximation of continuous functions
- Modulus of continuity
- Linear spline approximation
- Bernstein polynomials. Be able to define these and to show that
they span the appropriate space of polynomials.
- Weierstrass Approximation Theorem. Be able to sketch a proof,
given necessary properties of the Bernstein polynomials.
-
Fourier series
- Be able to find the Fourier series for a given function and to
use Parseval's equation to sum series or estimate $L^2$ errors.
- Riemann-Lebesgue Lemma. Be able to prove it.
- Pointwise convergence of Fourier series. Be able to sketch a proof of it.
- Given that $\{e^{inx}\}$ is a complete orthogonal set, be able to
use Parseval's inequality to sum a series.
Updated 10/1/2021 (fjn).