Math 642 Final Exam Review – Spring 2015

Operators on Hilbert space

• Unbounded operators
  • Terms: densely defined, closed & closable, extensions, adjoint, self adjoint, resolvent operator, resolvent set, spectrum (discrete, continuous, and residual)
• Resolvent operator (L − λI)⁻¹
  • Analytic, bounded on the resolvent set.
  • Spectrum of a self-adjoint operator — real; no residual spectrum.
• Spectral theorem
  • Be able to define the term spectral family, E_λ. Be able to state the spectral theorem.
  • Kodaira/Weyl/Stone formula, Green's functions, and spectral transform. Be able to find the spectral transform in simple cases — Fourier transform, Fourier sine and cosine transforms.

Fourier transforms

• Definition of transform and inverse transform. (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 and be able to establish simple L¹ properties of convolutions — f,g ∈ L¹ implies that f∗g ∈ L¹.
• Be able to prove or sketch a proof for each of these:
  • ``Useful'' form of Parseval's Theorem.   $\int_{\mathbb R} f(u)\hat g(u)du = \int_{\mathbb R} \hat f(u)g(u)du$.
  • Sampling theorem
  • Uncertainty principle

Schwartz space and tempered distributions

• Schwartz space S (See Feldman's online notes).
  • Definition and notation
  • Semi-norm and (equivalent) metric space topologies
  • Theorem. The Fourier transform of S is S. Be able to sketch a proof of this.
• Tempered distributions, S′ (See Feldman's online notes.)
  1. Definition and notation, derivatives of distributions, multiplcation of a distribution by C^∞ functions increasing polynomially
  2. Know the coninuity test for a linear functional to be a tempered distribution (Feldman, Theorem 6, pg. 7)
  3. The Fourier transform of a tempered distribution is defined via Parseval's identity,
     $\int_{\mathbb R} T(u)\hat \phi(u)du = \int_{\mathbb R} \hat T(u)\phi(u)du$
     
  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.
• Practice problems. See assignment 9.

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 6) 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 5/8/2015 (fjn).