Speaker: Matt Hirn, Emily King (UMD)

Title: Intro to Harmonic Analysis


This session will include 2 short, accessible talks on subjects related to harmonic analysis.

Uncertainty Principles for Finite Abelian Groups (Matt Hirn)

Let f be a continuous time function, i.e. f takes values from the real line into the complex plain. The classical uncertainty principle states that f and its Fourier transform cannot both be highly concentrated. In quantum mechanics the uncertainty principle shows that a particle’s position and momentum cannot be determined simultaneously (Heisenberg). Now let f be a discrete-time function taking elements from the group of integers mod N into the complex plain. We examine the corresponding discrete-time uncertainty principle for f, along with its generalization to any finite abelian group. Of particular note is the case when N is a prime number. We will find the discrete-time uncertainty principle can be improved in this case, leading to several interesting consequences. Work today that is rooted in the discrete-time uncertainty principle includes the representation of sparse signals in overcomplete dictionaries as well as compressed sensing.

A Casual Introduction to Frames (Emily King)

Frames can be viewed as generalizations of orthonormal bases. Given a Hilbert space, like $\mathbb{C}^d$, a frame is a collection of vectors which satisfy a weaker version of Parseval's identity. Frame theory has strong connections with operator theory and functional analysis, as well as harmonic analysis. Frames are also very useful in applications, from image processing to antenna theory. This talk will introduce the basics of frame theory with lots of examples and pictures. A knowledge of linear algebra is the only pre-requisite for this talk.