The midterm test (Friday, July 26) will have 5 to 7 questions, some
with multiple parts. Problems will be similar to ones done for
homework.
Fourier transforms
Definition of the FT on L^{1} and algebraic
properties. See sections 1.1.2, 1.2.1. There are also several
important spaces: L^{1}, L^{2}, L^{infinity},
BV, A, and spaces of continuous functions.
Definition of the FT on L^{2}. We approached the
problem differently than the text in that we started out by defining
the FT as an L^{2} limit of
F_{n} = FT[f 1_{[-n,n)}]
as n --> infinity. See Notes, 16 July. The formula we start with is
mentioned in section 1.10.14a.
Computing Fourier transforms. Be able to compute the
Fourier transforms of functions similar to ones dealt with in class
and in the homework assignments. Functions that we have dealt with
include the sinc or Dirichlet function, the Poisson
function, the Gaussian, and the Fejér
function.
Analytic properties of the FT. Section 1.4.1. You should
also be able to establish the simple properties listed on pg. 17 of
the text - namely, boundedness, time differentiation, and frequency
differentiation. You should know both the uniform continuity of the FT
of L^{1} functions, and be able to prove the Riemann-Lebesgue
Lemma.
Convolutions. Section 1.5. Be able to compute the
convolution of two functions. Be able to state and prove the
convolution theorem when f and g are in L^{1}.
Inversion formulae.
Pointwise inversion. The theorems we dealt with were Pringsheim's and
Jordan's. We actually proved an inversion formula that was similar to
Jordan's, but replaced BV with a ``Dini condition.'' See sections 1.1.6,
1.1.7, and Notes, 12 July. The nicest of the pointwise inversion
theorems is when f is both in L^{1} and A. See 1.7.8.
L^{2} inversion. Once one has established that the FT exists
for all f in L^{2}, and that it is also in L^{2}
(Notes, 16 July), we get that if F = FT[f], then f(t)=FT[F](-t).
A inversion. When F=FT[f] for f in L^{1}, then F belongs to
the space A. The inversion formula here is actually more of a
``recovery'' operation. The way to do this is to form
f_{m}(t) = FT[(1 - 2u π/m))_{+}F](-t).
As m --> infinity, || f_{m} - f ||_{L1}
--> 0. See section 1.6.9 and Notes, 12 July. This also shows that
the FT of an L^{1} function is unique.
L^{2} - the space of finite energy signals.
If f is in both L^{1} and L^{2}, then || F
||_{L2}=|| f ||_{L2}. This uses
the monotone convergence theorem (B. Levi).
Parseval/Plancheral Theorem: If f and g are in L^{2}, then
|| F ||_{L2}=|| f ||_{L2}
< f,g > = < F,G > (Usual inner product.)
Convolution Theorem: If f and g are in L^{2}, then f*g =
FT^{-1}[FG]. (Note: FG is in L^{1} and f*g is in A.)
Linear filters. Know what a linear, time-invariant
filter is, and what its impulse response function and system function
are. Be able to ``derive'' the structure theorem, L[f]=f*h. Be able to
define the term causal filter, and be able to do problems
similar to ones assigned for homework.
Fourier series
Definition for a 2W periodic L^{1} function F(w)
Fourier coefficients f[n] are regarded as a time series and F(w) the
frequency representation of the time series.
Formula for f[n].
Riemann-Lebesgue Lemma for FT applies here: f[n] -->0 as n -->
infinity.
Convergence results.
Partial sums S_{N}(w) are given as integrals of F against
the kernel D_{N}(t)=sin((N+½)t)/sin(½t).
Dini conditions for convergence of FS; proved using FT version of
results together with Dirichlet kernel.
Operations with Fourier series
Addition and scalar multiplication
Term-by-term integration (two theorems)
Term-by-term differentiation
L^{1} theory
Convolution theorem
Recovery of F from f[n].
Fejér Kernel W_{N} defined as the Cesáro
mean of D_{N}
F*W_{N} --> F (F continuous or L^{1})
Uniqueness theorem for Fourier coefficients of F in L^{1}
L^{2} theory
{exp(-iπn wW^{-1})} is an orthonormal basis for square
integrable 2W periodic functions
Parseval's formula
Riesz-Fischer theorem
Distribution Theory
Distributions
D=C_{c}^{infinity} is the space of
infinitely differentiable, compactly supported functions. Convergence
of a sequence {f_{n}} of functions in D to f in
D means that all members of the sequence are supported on a
compact set K and
|| f_{n}^{(k)} - f^{(k)} ||_{inf}
--> 0 as n --> infinity.
D' is the dual to D, and consists of all
continuous linear functionals on D; these are the
distributions. Examples: delta functions; locally integrable
functions. Here are some elementary properties of distributions
Linearity and continuity (definition)
Equality of distributions
Support of a distribution
Sum of distributions
Positive distributions
Derivatives of distributions: T' is defined via (T',f)=-(T,f').
Translation of distributions: (t_{u}T), f) := (T,t_{-u}f)
Tempered distributions
S, Schwartz space, is the space of infinitely
differentiable functions with ``fast'' decay. (See § 2.4.3.)
The Fourier transform of S is S.
S' consists of all the continuous linear functionals on
S. These are the tempered distributions. It turns
out that every tempered distribution is also a distribution. So the
properties listed above apply to tempered distributions, too.
The Fourier transform of a tempered distribution T is defined via
(FT[T],f) := (T,FT[f]). Note: the FT is not defined for all
distributions in D', just for those in S'.
Compactly supported distributions. The space of these is denoted
by E'.
Measures. For us, the most important class of measures
are the bounded Radon measures, M_{b}(R). This is the
set of all F' in D' such that F is in BV(R). (These
are also tempered distributions.) Know the definition. See
§2.3.6a.
Convolutions. One can find the convolution S*T of two
distributions S and T under the following circumstances.
S and T are measures in M_{b}(R). §2.5.2
S is in E' and T is D'. §2.5.4b
S and T are in S'. See §2.5.8a. Also, the convolution
theorem holds whenever S*T exists. See §2.5.9.
Positive definite functions & Bochner's theorem
Let f: R --> C be continuous. We say that f is
positive definite if, for every set of distinct points
{x_{1},...,x_{n}} in R, the n×n matrix A
with entries A_{j,k} = f(x_{j} - x_{k}) is
positive semidefinite.
Bochner's Theorem: P is positive definite if and only if P is the
FT (distribution sense) of a nonnegative measure µ in
M_{b}(R). Be able to prove that if µ is
nonnegative, then P=FT[µ] is a positive definite function.