# Research

## Analysis (Leader: Prof. Dmitry Khavinson <dkhavins (at) usf.edu>document.write('<a href="mai' + 'lto:' + 'dkhavins' + '&#64;' + 'usf.edu' + '">Prof. Dmitry Khavinson</a>');)

### Friday, December 3, 2021

Title: Orthogonal polynomials on the real axis, Part II
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130 (Hybrid Format)

### Friday, November 26, 2021

No seminar this week due to the Thanksgiving Holiday.

### Friday, November 19, 2021

Title: Orthogonal polynomials on the real axis
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130 (Hybrid Format)

#### Abstract

I plan to make a few general remarks on history and current state of the art and then discuss in some details a couple of particular problem related to asymptotics.

### Friday, November 12, 2021

Title: On a localization result in billiard systems of elliptical shape
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: CMC 130 (Hybrid Format)

#### Abstract

In their “theory of quantum mirage” (2001), Oded Agam and Avraham Schiller computed perturbatively the resolvent for the generator of diffusion in a “quantum billiard” of elliptical shape. The resulting probability distribution function has singleton contributions at the focal points of the ellipse, independently of its size or aspect ratio. We will discuss a mathematical formulation of the Agam-Schiller result.

### Friday, November 5, 2021

Title: Commutator-preserving deformations of the universal algebra of the space of square-summable sequences, Part II
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: CMC 130 (Hybrid Format)

### Friday, October 29, 2021

Title: Commutator-preserving deformations of the universal algebra of the space of square-summable sequences
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: CMC 130 (Hybrid Format)

#### Abstract

Starting from the pair of shift operator, $$W$$, on $$\ell^2(\mathbb{N})$$ and its adjoint $$W^*$$, we construct the simplest pair of operators with identity commutator, $$Z$$ and $$\bar{Z}$$. They are represented by multiplication and differentiation operators acting on the Segal-Bargmann space on $$\mathbb{C}$$, leading to an orthonormal basis of monomials with an obvious one-parameter (L) group of scaling invariance (the Moebius subgroup leaving 0 and oo invariant). Correspondingly, the commutator of $$Z,\bar{Z}$$ scales as $$|L|^2$$.

We consider continuous deformations of the pair $$Z,\bar{Z}$$ into the algebra of formal Laurent series in $$W,W^*$$, leaving the commutator invariant. We show that the simplest case leads to an orthonormal basis of (complex) Hermite polynomials, with deformed inner product, and we explore the asymptotic behavior of the projector on the n-th polynomial, using the residual scaling invariance. An unexpected by-product is a theorem showing that diffusion paths starting from focal points of elliptical domains visit the focal points infinitely often.

Higher-order deformations will also be discussed if time allows.

### Friday, October 22, 2021

Title: On a boundary property of Blaschke products
Speaker: Arthur Danielyan
Time: 4:00pm–5:00pm
Place: CMC 130 (Hybrid Format)

#### Abstract

We prove that a Blaschke product has no radial limits on a subset $$E$$ of the unit circle $$T$$ but has unrestricted limit at each point of $$T \backslash E$$ if and only if $$E$$ is a closed set of measure zero. (Joint work with Spyros Pasias).

### Friday, October 15, 2021

Title: Pseudoinverses in $$H^2\left(\mathbb{D}^d\right)$$, Part II
Speaker: Ray Centner
Time: 4:00pm–5:00pm
Place: Microsoft Teams

### Friday, October 8, 2021

Title: Pseudoinverses in $$H^2\left(\mathbb{D}^d\right)$$
Speaker: Ray Centner
Time: 4:00pm–5:00pm
Place: Microsoft Teams

#### Abstract

The methods of least-squares approximation have been used in many areas of engineering since the later half of the 20th century. In particular, these methods appear in connection with 2D recursive digital filter design. In this context, electrical engineers are concerned with manipulating polynomials of two complex variables in an effort to control the output of a 2D signal. However, difficulties arise in studying these techniques due to limited factorization results of polynomials in several variables. In effort to avoid this problem, mathematicians have been interested in approaching this topic from a more abstract point-of-view. In this talk, I will present a method to studying these polynomials from an operator theoretic standpoint. In particular, I will talk about the pseudoinverse of an operator that is defined in conjunction with these polynomials. I will then show how some previously known results can be obtained easily by using this framework. To maintain flexibility with the engineering applications, I will present a majority of the results in the $$d$$-dimensional case.

### Friday, October 1, 2021

Title: The classification problem for arclength null quadrature domains, III
Speaker: Dima Khavinson
Time: 4:00pm–5:00pm
Place: Microsoft Teams

### Friday, September 24, 2021

Title: The classification problem for arclength null quadrature domains, II
Speaker: Dima Khavinson
Time: 4:00pm–5:00pm
Place: Microsoft Teams

### Friday, September 17, 2021

Title: The classification problem for arclength null quadrature domains
Speaker: Dima Khavinson
Time: 4:00pm–5:00pm
Place: Microsoft Teams

#### Abstract

A planar domain is referred to as an arclength null quadrature domain if the integral along the boundary of any analytic in the domain function, representable by the Cauchy integral of its boundary values (Smirnov class $$E^1$$), vanishes. Obviously, all such domains must be unbounded. We prove the existence of a “roof function” (a positive harmonic function whose gradient coincides with the inward pointing unit normal along the boundary) for arclength null quadrature domains having finitely many boundary components. This bridges a gap toward classification of arclength null quadrature domains by removing an a priori assumption from previous classification results, in particular, it completes the theorem of Khavinson-Lundberg-Teodorescu from 2012 that established that domains allowing “roof functions” are arclength null quadrature domains. This result also strengthens an existing connection to free boundary problems for Laplace's equation and the hollow vortex problem in fluid dynamics. The proof is based on the techniques originated in classical works of Ahlfors, Carleman and Denjoy. We shall also discuss the current status of the classification problem for arclength null quadrature domains. This is a 2021 joint work with Erik Lundberg.