Math 447 – Advanced Analysis I
Review for Midterm Exam
The exam will consist in 2 theoretical questions and 3 problems similar to the homework problems. One page of notes is allowed, but not containing solutions to homework problems.
Ch.1 Measure and Integration
-sigma-algebra, measurable function, measurable space, products of sigma-algebras,
-Borel measurable sets and functions, compatibility with algebraic operations and point-wise limits.
-Simple functions, density of simple functions within measurable functions.
-Positive measures: definition, properties, sets of measure zero
-Inner/outer measure of a finite measure, Lebesgue measurable sets, translation invariance
-Stieltjes measure, probability spaces, measure zero sets
-Integral of measurable functions: integral of simple functions, extension by continuity
-Construction of the integral as an area (product measure): inner measure = outer measure
-The Riemann integral (simple functions) and Lebesgue integral as an extension
-L1-space and L1/~ measure zero quotient, with norm defined as an integral.
-Probability spaces, conditional probability
-Lebesgue Dominated Convergence theorem: pointwise convergence + uniform domination by an integrable function => convergence in norm; => lim Int=Int lim by continuity of Int.
-Applications of LDCTh: integrals depending on a parameter & continuity/differentiability
-Fubini’s Theorem: product measures, Fubini’s Theorem for measures (existence of the product measure), the double integral and the product measure, the iterated triple product measure and Fubini’s Theorem.
-The convolution product: definition, domain, properties.
Ch. 2 Lp-Spaces
-Inequalities: Minkowski, Holder, Jensen.
-Lp-spaces: definition, norm, distance; L2 as a Hilbert space;
-Completeness: Riesz-Fisher Th., compact support, finite support, dense subspaces
-Fourier transform: def., properties, Fourier inversion formula
-Applications to probabilities and differential equation: random variables, harmonic functions.
-Laplace transform: def., domain, properties, the convolution theorem, application to DE.
Ch. 3 Hilbert Spaces
-Scalar product, quadratic form/norm, polarization, Cauchy-Schwarz, orthogonal complement, Pythagoras Theorem, Minkowski inequality, the parallelogram law.
-Hilbert space, orthonormal projection theorem, orthogonal projector, direct sum of Hilbert spaces
-Projections, orthogonal projections, Riesz Theorem
-Orthogonal systems, complete ON-systems (Hilbert basis), Graham-Schmidt ON-process, examples of orthogonal polynomials
-Review of Fourier series in L2: basis of characters, Parseval’s Theorem, Plancherel Theorem (characters form an ONB).
-Pointwise convergence of Fourier series: smooth case or continuous case: Dirichlet’s kernel (partial sums), Cesaro means & Fejer’ skernel, Fejer’s Theorem.
Ch.4 Distributions
-Test functions, distributions, regular and singular distributions
-Differentiation of distributions, as an extension of d/dx via embedding functions in D’, product rule, Heaviside function’=delta
-Partial product of distributions, transformations of distributions, invariant distributions
-Convergence of distributions, derivative is continuous, Dirac sequences
-Convolution: definition, domain and convolution algebra, support of distributions, properties
-Applications to DE and integral equations -> convolution equations.
-Fourier transform and Laplace transform of distributions: definition, properties and applications.
Exercises posted on the Web
Ch.1
- Exercise 1: Prove that the counting measure, and the Dirac measures (unit measure concentrated at a point) d_a(A)=Char_A(a) are indeed measures.
- Exercise 2: a) Borel sets are measurable; b) Not all sets are measurable: a section" of [0,1]/Q is not measurable.
- Exercise 3: Lebesgue measure is translation invariant.
- Exercise 4: Prove that if f(x) is discontinuous at a, then the associated Stieltjes measure satisfies the condition m_f({x})=f(x+0)-f(x).
- Exerice 5. Assuming that (X,P) is a probability space, prove Bayes' formula: P(Hi|B)=P(Hi) P(B|Hi)/P(B), where P(A|B)=P(A B)/P(B) is the conditional probability of A given the event B.
- Exercise 6: a) If sA is a sigma algebra and A is a set of sA, then sA&A is a sigma algebra; b) If m is a measure and m_A(B)=m(A&B), then m_A is a measure, called the restriction of m to A.
- Exercise 7. Prove that a countable set of real numbers is Lebesgue negligible.
- Exercise 8: Prove that f(x;m_A)=m_A(-infinity, x] is continuous if A is measurable of finite Lebesgue measure.
- Exercise 9: f=g a.e. is an equivalence relation.
- Exercise 10: If f(x) is integrable => {x|f(x)=infinity} is m-negligible.
- Exercise 11: Prove that {f<L1| ||f||=0} is a vector subspace of L1(X,m).
Ch. 2
- Exercise 1 Given the Bessel function J0(x)=1/pi Int(cos(x sin u) du, 0, pi) use the intertwining properties of LT to prove that L(z, J0)=1/(1+z^2)^(1/2).
- Exercise 2 Solve the differential equation x y'' + 2 y' + xy=0 by applying the Laplace Transfiorm and using its intertwining properties.
- Exercise 3: (Example 7) Solve the integral equation
f(x) - ([k(x) 1_(0,x)]* f)(x)=g(x)
using the properties of the LT.
Ch. 3
Exercise 1: Prove that the scalar product is given in terms of the associated norm, by the following polarization formula: …
Exercise 2: Use the quadratic formula to prove the Cauchy-Schwarz inequality |<x|y>|<= ||x|| ||y||.
Exercise 3: Prove the parallelogram formula ||x+y||2+||x-y||2=2||x||2+2||y||2.
Ch. 4
Exercise 1: Prove that Tf Tg=T(fg), so that embedding functions as regular distributions is an algebra morphism.
Exercise 2: If u:Rn->Rn linear, invertible, a) then u(T)()=|J| T(u-1) is a distribution, where u^(-1)j(x)=j(u(x)); b)
Exercise 3: If u:Rn->Rn is linear invertible transformation and u(T)()=|J| T(u-1) denotes the transformed distribution, then for regular distributions u(Tf)=T(uf).
Exercise 4: Prove that differentiation of distributions is a continuous operator.
Exercise 5: Prove that 1) * T=T, 2) ‘* T=T’.
Exercise 6: Prove that (S * T)’=S * T’=S’ * T.
Math 447 – Advanced Analysis
Review for Final Exam
Final exam will consist of 5 problems: 2 homework problems and 3 theoretical questions (stating definitions and properties of the main concepts).
One page of notes is allowed, but without solutions to homework problems. Review the theory for each section and the corresponding homework problems.
Ch. 5 Linear Operators
-Def. linear operator, bounded linear operator, norm, convergence
-The Hahn-Banach Theorem
-Hermitean operators and the adjoint
-Unitary and self-adjoint operators
-Projectors
-Isometries
-Compact operators: def., properties: form a closed algebra
-Unbounded operators, example: p & q
-Spectral Theory: resolvent, spectrum, spectral radius
-Spectral Theorem for Compact Operators: A=Sum cn Pn; E=ker A + Sum En and resolution of identity.
Part II – Introduction to Complex Analysis
-Holomorphic functions, Cauchy-Riemann equations, Cauchy Theorem, Cauchy Formula
-Connected, simply connected
-Lourent series and Laurent Theorem, singularities (classification and behavior)
-Meromorphic functions, Riemann sphere (extended complex plane), differentiability at infinity
-Multifunctions, branches and cuts: the exponential and logarithm example
-Residue, Cauchy Residue Theorem
-Counting zeros and poles, Rouche Theorem, Corollary: Fundamental Theorem of Calculus
-Computing the residue
-Indentations and estimating contour integrals on arcs
-Applications: defining and evaluating improper integrals and principal value integrals, using contour complex integration.
Problems
1) Let P=d/dx and Q=x (i.e. (Qf)(x)=x f(x)), be the momentum and position operators defined in L2(R). Compute their commutator.
2) Prove that the anti-derivative operator defined on C[a,b], is bounded with respect to the sup norm, and its norm is ||R||=(b-a).
3) Prove the Hahn-Banach Theorem.
4) Prove that a unitary operator is an isometry.
5) Prove that a bounded operator on a finite dimensional Hilbert space is compact.
6) The annihilation operator on l2(N): Ae0=0, Ae(n)=sqrt(n) e(n-1).
a) Prove that A is unbounded and that the domain is dense in l2(N).
b) Prove that its adjoint is A*(en)=sqrt(n+1) e(n+1) (the creation operator).
c) Compute their commutator: [A,A*].
7) Derive the Cauchy-Riemann equations satisfied by a holomorphic function.
8) Prove that the real and imaginary parts of a holomorphic function satisfies Laplace equation.
9) State the Cauchy theorem.
10) State the Deformation Theorem.
11) Prove that a function satisfies Cauchy Theorem iff it satisfies Cauchy Theorem for simply connected domains.
12) Evaluate the following contour integrals, where C(a;r) denotes a circle centered at a of radius r:
,
13) State Laurent’s Theorem.
14) Compute the Laurent series of f(z)=1/(z (z-1)) about z=0.
15) Classify the zeros of f(z)=z sin(z).
16) Clasify the poles of z tan(z).
17) Prove that f(z)=1/z is differentiable at infinity.
18) Determine and classify the poles of 1/(z2+1) and compute their residues.
19) Compute the principal value of 1/(x2+1), using contour integration.