| Date
|
Speaker
|
Title
|
| 22 January |
Eugenio Hernández (Universidad Autónoma de Madrid) |
Democracy for a collection of translates of a function |
| Abstract: The democratic property of a basis in a Banach space is closely related to the near-best approximation property of the basis for N-term non-linear approximation. Also, principal shift invariant subspaces in L2(R) are central in the theory of wavelets. These are generated by the integer translates of a function in L2(R). We will study under which conditions the system of integer translates of a function is a democratic collection in L2(R). The problem is still unsolved.
|
| 29 January |
Brody Johnson (SLU) |
Nonlinear approximation with dual frames |
| Abstract: This talk will describe some basic observations about the performance of a greedy algorithm for the N-term nonlinear approximation problem in a Hilbert space using a pair of dual frames. These results will be contrasted to the notion of a greedy basis as described in the January 22 seminar.
|
| 5 February |
Brody Johnson (SLU) |
Nonlinear approximation with dual frames |
| Abstract: Continuation of the January 29 seminar.
|
| 12 February |
No Seminar |
|
| Abstract:
|
| 19 February |
Chun-Yen Shen
(Indiana University) |
Explicit sum-product estimates of different sets in finite fields |
| Abstract: The sum-product phenomenon has received a great deal of attention, since Erd\"{o}s and Szemer\`{e}di made their well known conjecture that $\max(|A+A|,|AA|) \geq C_{\epsilon} |A|^{2-\epsilon} \forall \epsilon > 0.$ where $A$ is a finite subset of integers and $A+A=\{a+b: a \in A, b \in A \},$ and $AA=\{ab: a \in A, b \in A \}.$ In this talk ,we will discuss the analogy results in finite fields and its applications. In particular, we address how to use Garaev's inequalities to get quantitative sum-product estimates in finite fields and how Fourier analysis could be applied to attack these kinds of problems.
|
| 26 February |
Tom McNamara (SLU) |
Group Representations and Special Functions |
| Abstract: Special functions might be described as "the functions arising from the mathematical study of physical problems". Many of these physical problems are modelled using differential equations. We will look at several examples of such differential equations and show how Lie Theory helps unify apparently ad-hoc methods of solution.
|
| 4 March |
No Seminar |
|
| Abstract:
|
| 11 March |
No Seminar |
|
| Abstract:
|
| 18 March |
No Seminar |
Spring Break |
| Abstract:
|
| 25 March |
Tom McNamara (SLU) |
Classical Differential Equations and Lie Theory |
| Abstract: We will demonstrate through the use of several concrete examples the connection between Lie algebras and classical differential
equations. As a starting point, we show how the Rodrigues formula for the Legendre polynomials can be derived. We will also produce Rodrigues-type formulas for the Laguerre and Hermite functions. Further, we show how recursion relations and other properties of these functions can be derived in a formal manner.
|
| 1 April |
No Seminar |
|
| Abstract:
|
| 8 April |
Brody Johnson (SLU) |
Greedy iff Unconditional & Democratic, Part 1 |
| Abstract: Konyagin and Temlyakov showed that a basis for a Banach space is greedy if and only if it is unconditional and democratic. This talk will begin an exposition of their proof. Little familiarity with the notions of unconditional, greedy, or democratic bases will be assumed.
|
| 15 April |
Brody Johnson (SLU) |
Greedy iff Unconditional & Democratic, Part 2 |
| Abstract: Continuation of the April 8 seminar.
|
| 22 April |
Ashley Moses (SLU) |
Constructing Compactly Supported Wavelets |
| Abstract:
|
| 29 April |
Darrin Speegle (SLU) |
Dilations of Parseval frames generated by groups |
| Abstract:
Given a Parseval frame $\{x_i: i\in I\}$ for a Hilbert space $H$,
there exists a Hilbert space $H_1$ containing $H$ as a closed subspace and
an ONB $\{e_i: i\in I\}$ for $H_1$ such that $Pe_i = x_i$, where $P$ is the
orthogonal projection onto $H$. In this introductory talk, we show that if
the Parseval frame is generated by a group of unitary operators acting on a
single vector, then the ONB can also be generated by unitary operators on
$H_1$ representing the same group. Extensions to the Gabor and wavelet
case will also be considered. (This talk is based on my reading of a recent
paper by Dutkay, Han, Picioroaga and Sun.)
|