Functional Analysis, Harmonic Analysis/Fourier Analysis

Recommended Texts or Study Materials

Functional Analysis

  • W Rudin: Real and Complex Analysis Dr. Shipman says, "This is an excellent classic with a breadth of basics on measure and integration, abstract function spaces, functional analysis, and complex variables."
  • B V Limaye:  Functional Analysis
  • Topological Vector Spaces, Distributions and Kernels, François Treves.
  • A Course on Topological Vector Spaces, Jurgen Voigt.

Harmonic Analysis

  • E Stein: Singular Integrals and Differentiability Properties of Functions
  • G Folland: A Course in Abstract Harmonic Analysis

Foundational Tools/Techniques/Ideas:

Topics

  • Distribution theory
  • The open mapping theorem, closed graph theorem, and uniform boundedness theorem are fundamental to all areas of analysis; they occur repeatedly in abstract analysis.
  • Fourier analysis
  • Sequences and Series in different topological vector spaces (TVS) Note: this is a bit of a specialized area and not required of every analysis student.

Ideas

  • Harmonic analysis and Fourier analysis are intimately related, often being considered to be the same subject.
  • Abstract harmonic analysis is based on characters of group actions; when those groups become non-commutative, characters are replaced by more general representation theory.
  • Observe results true in finite-dimensional TVS and then prove whether they hold for a special class of infinite-dimensional TVS.
  • Use distributional integration instead of the usual Riemann/Lebesgue definition.

Common Advisors and Committee Members

Advisor Committee Member 1 Committee Member 2
Dr. Ricardo Estrada Dr. Stephen Shipman Dr. Padmanabhan Sundar

Analysis students give presentations, below are quotes from graduate students about that process.

Question: Did you present a paper or an idea?
  • "I presented a survey for the most part; followed with results, few as they may be, that we had obtained. The idea was to present stuff that would help me better prepare for research."
Question: What was the process for selecting your presentation material?
  • "We found an area that both I and my advisor d that we had questions about. We decided to study some papers in the vein of those questions. This gave us a good idea of what was known and what could be proven but hadn't yet been. My presentation was essentially a neat summary of the work done so far in answering those questions and then some results that we had managed to prove based on the work we had read."