Math 414 - Test I Review
General Information
Test I (Tuesday, March 5) will have 5 to 7 questions, some with
multiple parts. Please bring an 8½×11
bluebook. Problems will be similar to ones done for
homework. You may use calculators to do arithmetic, although you will
not need them. You may not use any calculator that has the
capability of doing either calculus or linear algebra.
Topics Covered
Inner Product Spaces
- Basics
- Definition of an inner product and its associated norm. Know
these well enough to be able to apply them. §0.2
- Schwarz and triangle inequalities. Be able to state and apply the
Schwarz and triangle inequalities. Be able to compute the angle
between two vectors. §0.4
- Linear transformations and adjoints. Be able to do problems
similar to ones for homework. Be able to show that an orthogonal
projection is self-adjoint. §0.6.1, §0.6.2
- L2 and types of convergence
- Know the definition of L2 and the inner product on
L2. §0.3.1.
- Convergence. For sequences of functions, we studied three types
of convergence: mean, uniform, and pointwise. Know the definitions of
these and be able to do examples similar to ones in the book,
homework, and class notes. §0.3.2.
- Orthogonality
- Orthogonal sets of vectors, orthonormal bases, and orthogonal
spaces. Know the definitions for these terms. Know how to write a
vector in terms of an orthonormal basis, and how to calculate the
coefficients. Be able to do problems similar to ones assigned in
homework. §0.5.1.
- Orthogonal projections. Know the definition of an orthogonal
projection. Be able to do problems similar to ones in the homework. Be
able to show that if uV is the orthogonal projection of u
onto V, then u - uV is orthogonal to V. §0.5.2.
- Applications. Least squares and linear predictive coding. Be able
to do a simple least-squares fit. Be able to describe the algorithm
used in linear predictive coding. §0.7.
Fourier Series
- Calculating Fourier Series
- Fourier series. Given a function, you should be able to compute a
Fourier series in either real or complex form, and with prescribed
period 2a.
- Fourier sine series and Fourier cosine series. Be able to compute
FSS and FCS for functions defined on a half interval, [0,a].
- Be able to use symmetry properties to help compute coefficients in
FS, FSS, FSC.
- Pointwise convergence
- Riemann-Lebesgue Lemma. Be able to give a proof of this in the
simple case that f is continuously differentiable. §1.3.1.
- Fourier (Dirichlet) kernel, P. Know what P is and how to express
partial sums in terms of P. §1.3.2
- Be able to sketch a proof of Theorem 1.22, making use of the
formula for P and the properties of P as well as the Riemann-Lesbegue
Lemma. §1.3.2.
- Know the Theorems 1.22 and 1.28. Be able to use them to decide
what function an FS, FSS, or FCS converges to.
- Uniform convergence
- Conditions under which an FS, FSS, or FCS is uniformly
convergent. Be able to apply these to determine whether or not an FS
is uniformly convergent. These are all stated for periodic
functions. To apply them on [-a,a], or [0,a], work with the
appropriate periodic extension. These conditions are not
equivalent. §1.3.4.
- SUMk (|ak| + |bk|) <
inf. (This can be used to estimate truncation error.)
- The function f is continuous, periodic, and has a piecewise
continuous derivative.
- Uniform convergence on subintervals. Let f be 2a periodic. If f
is continuous and has a piecewise continuous derivative on [A,B],
-a<A<B<a, then the FS for f is uniformly convergent on any
subinterval [A',B'], where A<A'<B'<B
- Gibbs' phenomenon. Be able to briefly describe the Gibbs'
phenomenon, and to discuss its universality.
- Be able to do problems similar to ones assigned for homework.
- Mean convergence
- Best approximation property of partial sums. Be able to show
Lemma 1.34, given that VN has an orthogonal basis. (The
partial sum is an orthogonal projection of f onto VN.)
§1.3.5.
- Parseval's theorem. Know both the real and complex form. be able
to use it to sum series similar to ones given in the homework.
- Here are the main theorems on mean convergence. Be able to state
them and briefly explain their significance.
- If f is in L2, then the partial
sums of the FS for f converge in the mean to f.
- Riesz-Fisher theorem. If SUMk
(|ak|2 + |bk|2) < inf,
then there exists an f in L2 such that the series formed
from the ak's and bk's is the FS for f and
converges in the mean to f.