Classical Analysis
(Leader: )
Friday, April 9, 2021
On Mergelyan sets and Farrell sets
Arthur Danielyan
4:00pm–5:00pm
MS Teams
Abstract
The concepts of Mergelyan sets and Farrell sets have been introduced by Lee Rubel in 1973 for his known joint approximation problem in complex domain and studied by many authors. We present the relevant classical results and consider some questions on Mergelyan sets and Farrell sets for \(Hp\) (\(1 \le p < \infty\)) spaces of analytic functions in the unit disc for both the weak topology and the norm topology.
Friday, April 2, 2021
Optimal Polynomial Approximants and Design of Digital Filters, Part II
Catherine Bénéteau
4:00pm–5:00pm
MS Teams
Friday, March 26, 2021
Optimal Polynomial Approximants and Design of Digital Filters
Catherine Bénéteau
4:00pm–5:00pm
MS Teams
Abstract
In this talk, I will discuss the notion of optimal polynomial approximants, which are polynomials that approximate, in some sense to be discussed, inverses of functions in certain Hilbert spaces of analytic functions. In the last 10 years, a number of papers have appeared examining the zeros of these polynomials, rates of convergence, efficient algorithms for their computation, and connections to orthogonal polynomials and reproducing kernels, among other topics. On the other hand, in the ‘70s, researchers in engineering and applied mathematics introduced least squares inverses in the context of digital filters in signal processing. It turns out that in the Hardy space \(H^2\) the optimal polynomial approximants and the least squares inverses are identical. In this talk, I will survey known results and some open problems related to the zeros of optimal polynomial approximants and implications for the design of ideal digital filters. This talk is based on a survey paper that is joint with Ray Centner. Don't worry, I will start from the beginning.
Friday, March 19, 2021
No seminar this week.
Friday, March 12, 2021
Polynomial interpolation on arbitrary varieties, Part II
Boris Shekhtman
4:00pm–5:00pm
MS Teams
Friday, March 5, 2021
Polynomial interpolation on arbitrary varieties
Boris Shekhtman
4:00pm–5:00pm
MS Teams
Abstract
I will talk about the following problem: Let \(V_1,\dotsc,V_{n}\) be varieties in \(\mathbb{C}^{d}\) and let \(p_1,\dotsc,p_{n}\) be given polynomials of \(d\) variables. When can we find one polynomial \(p\) such that \(p=p_{j}\) on each variety \(V_{j}\)? This, to the best of my knowledge, is the first extension of classical interpolation problem when \(V_{j}\) are chosen to be points. All results, in my opinion, are cute and proofs are very simple.
Friday, February 26, 2021
Quadratic differentials and associated extremal problems, Part III
E. Rakhmanov
4:00pm–5:00pm
MS Teams
Friday, February 19, 2021
Quadratic differentials and associated extremal problems, Part II
E. Rakhmanov
4:00pm–5:00pm
MS Teams
Friday, February 12, 2021
Quadratic differentials and associated extremal problems
E. Rakhmanov
4:00pm–5:00pm
MS Teams
Abstract
It is well known that each rational guadratic differential on the sphere without recurrent trajectories solves a number of extremal problems which may be formulated in different terms: electrostatic, metric or other.
I will discuss a few examples having in mind to outline some kind of classification for problems associated with a closed QD. It is known that it is not possible. (Too many problems as the last talks confirmed.) Anyway ...
Friday, February 5, 2021
Finding solutions to a free boundary problem involving surface tension, Part III
Nathan Hayford
4:00pm–5:00pm
MS Teams
Friday, January 29, 2021
Finding solutions to a free boundary problem involving surface tension, Part II
Nathan Hayford
4:00pm–5:00pm
MS Teams
Friday, January 22, 2021
Finding solutions to a free boundary problem involving surface tension
Fudong Wang
4:00pm–5:00pm
MS Teams
Abstract
We will discuss how to construct explicit solutions to a free boundary problem considered by E. B. McLeod (1955), and later by P. R. Garabedian (1965). A one-parameter family of droplets to a free boundary problem with prescribed poles will be investigated following the scheme introduced in the paper by Khavinson-Solynin-Vassilev (2005). This is joint work with N. Hayford.
Friday, January 15, 2021
Why \((2,3)\)-cusp is impossible in Hele-Shaw droplets
Seung-Yeop Lee
4:00pm–5:00pm
MS Teams
Abstract
Sakai classified all the possible cusp singularities that can happen in Hele-Shaw droplets. We will go over the proof.