Math 658 - Midterm Review

General Information

Fourier transforms

• Definition of the FT on L¹ and algebraic properties. See sections 1.1.2, 1.2.1. There are also several important spaces: L¹, L², L^infinity, BV, A, and spaces of continuous functions.

• Definition of the FT on L². We approached the problem differently than the text in that we started out by defining the FT as an L² 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 L¹ 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 L¹.

• 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<sup>1</sup> and A. See 1.7.8.
  • L<sup>2</sup> inversion. Once one has established that the FT exists for all f in L<sup>2</sup>, and that it is also in L<sup>2</sup> (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<sup>1</sup>, 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 L¹ function is unique.

• L² - the space of finite energy signals.
  • If f is in both L¹ and L², then || F ||_L²=|| f ||_L². This uses the monotone convergence theorem (B. Levi).
  • Parseval/Plancheral Theorem: If f and g are in L², then
    • || F ||_L²=|| f ||_L²
    • < f,g > = < F,G > (Usual inner product.)
  • Convolution Theorem: If f and g are in L², then f*g = FT^-1[FG]. (Note: FG is in L¹ 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 L¹ 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
• L¹ 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 L¹)
    • Uniqueness theorem for Fourier coefficients of F in L¹
• L² 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_uT), f) := (T,t_-uf)
• 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 {x₁,...,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.