Math 642 Final Exam Review – Spring 2011

Operators on Hilbert space

• Unbounded operators (See notes, 3/21/11, 3/23/11.)
  • Terms: densely defined, closed & closable, extensions, adjoint, self adjoint.
  • Be able to show that L* is closed.
• Spectrum of an operator L, σ(L)
  • Discrete, continuous, residual. Know that σ(L) is a closed set.
  • Resolvent set of L, ρ(L). Know that ρ(L) is an open set.
• Resolvent operator (L − λI)⁻¹
  • 1st resolvent identity (notes, 3/28/11). Be able to state and use this.
  • Analytic, bounded on the resolvent set.
  • Spectrum of a self-adjoint operator — real; no residual spectrum.
  • Connection with projections (notes, 4/1/11).
• Spectral theorem
  • Be able to define the term spectral family, E_λ. Be able to state the spectral theorem (notes, 4/4/11, 4/6/11)
  • Kodaira/Weyl/Stone formula, Green's functions, and spectral transform. Be able to find the spectral transform in simple cases (notes, 4/8/11, 4/11/11).

Fourier transforms

• Definition of transform and inverse transform (notes, 4/13/11, 4/15/11; section 7.2.1). (Use whichever sign convention you want, just be consistent.)
• Be able to establish simple properties (Theorem 7.2).
• Be able to compute transforms and inverses of transforms. Know the convolution theorem.
• ``Useful'' form of Parseval's Theorem.   ∫_R f(u)g^(u)du = ∫_R f^(u)g(u)du (f^ and g^ are the Fourier transforms of f and g.)
• Uncertainty principle. Be able to prove this.

Schwartz space and tempered distributions

• Schwartz space S (See Feldman's online notes).
  1. Definition and notation
  2. Semi-norm and (equivalent) metric space topologies
  3. Theorem. The Fourier transform of S is S. Be able to sketch a proof of this (notes, 4/22/11).
• Tempered distributions, S′ (See Feldman's online notes.)
  1. Definition and notation
  2. Derivatives multiples of distributions
  3. The Fourier transform of a tempered distribution is defined via Parseval's identity,
     ∫_R T(u)f^(u)du = ∫_R T^(u)f(u)du
     
  4. Examples of Fourier transforms of tempered distributions

Asymptotic analysis

• Asymptotic estimates and series; big "O" and little "o" notation
• Watson's lemma. Be able to state, prove, and use it.
• Laplace's method. Be able to derive asymptotic estimates, e.g. Stirling's formula.
• Examples. See assignment 7.

Structure of the exam

There will be 5 to 7 questions. You will be asked to state a few definitions, and to do problems similar to assigned homework problems (starting with assignment 5) and examples done in class. In addition, you will be asked to give or sketch a proof for a major theorem or lemma from the material above.

Updated 5/4/2011 (fjn).