## Seminars and Colloquia by Series

### Agler Decompositions on the Bidisk

Series
Analysis Seminar
Time
Wednesday, April 18, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kelly BickelWashington University - St. Louis
It is well-known that every Schur function on the bidisk can be written as a sum involving two positive semidefinite kernels. Such decompositions, called Agler decompositions, have been used to answer interpolation questions on the bidisk as well as to derive the transfer function realization of Schur functions used in systems theory. The original arguments for the existence of such Agler decompositions were nonconstructive and the structure of these decompositions has remained quite mysterious. In this talk, we will discuss an elementary proof of the existence of Agler decompositions on the bidisk, which is constructive for inner functions. We will use this proof as a springboard to examine the structure of such decompositions and properties of their associated reproducing kernel Hilbert spaces.

### The s-Riesz transform of an s-dimensional measure in R^2 is unbounded for 1<s<2

Series
Analysis Seminar
Time
Wednesday, April 11, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
This is a joint work with F.~Nazarov and A.~Volberg.Let $s\in(1,2)$, and let $\mu$ be a finite positive Borel measure in $\mathbb R^2$ with $\mathcal H^s(\supp\mu)<+\infty$. We prove that if the lower $s$-density of $\mu$ is+equal to zero $\mu$-a.~e. in $\mathbb R^2$, then$\|R\mu\|_{L^\infty(m_2)}=\infty$, where $R\mu=\mu\ast\frac{x}{|x|^{s+1}}$ and $m_2$ is the Lebesque measure in $\mathbb R^2$. Combined with known results of Prat and+Vihtil\"a, this shows that for any noninteger $s\in(0,2)$ and any finite positive Borel measure in $\mathbb R^2$ with $\mathcal H^s(\supp\mu)<+\infty$, we have+$\|R\mu\|_{L^\infty(m_2)}=\infty$.Also I will tell about the resent result of Ben Jaye, as well as about open problems.

### Truncated Toeplitz operators

Series
Analysis Seminar
Time
Monday, March 26, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Dan TimotinIndiana University and Mathematical Institute of Romania
Truncated Toeplitz operators, introduced in full generality by Sarason a few years ago, are compressions of multiplication operators on H^2 to subspaces invariant to the adjoint of the shift. The talk will survey this newly developing area, presenting several of the basic results and highlighting some intriguing open questions.

### Optimal error estimates in operator-norm approximations of some semi-groups

Series
Analysis Seminar
Time
Wednesday, March 14, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vygantas PaulauskasVilnius University
In the talk some problems related with the famous Chernoff square root of n - lemma in the theory of approximation of some semi-groups of operators will be discussed. We present some optimal bounds in these approximations (one of them is Euler approximation) and two new classes of operators, generalizing sectorial and quasi-sectorial operators will be introduced. The talk is based on two papers [V. Bentkus and V. Paulauskas, Letters in Math. Physics, 68, (2004), 131-138] and [V. Paulauskas, J. Functional Anal., 262, (2012), 2074-2099]

### Rearrangements of Fourier Series

Series
Analysis Seminar
Time
Wednesday, March 7, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mark LewkoUniversity of Texas
We will discuss several results (and open problems) related to rearrangements of Fourier series, particularly quantitative questions about maximal and variational operators. For instance, we show that the canonical ordering of the trigonometric system is not optimal for certain problems in this setting. Connections with analytic number theory will also be given. This is based on joint work with Allison Lewko.

### Recent asymptotic expansions related to numerical integration and orthogonal polynomial expansions

Series
Analysis Seminar
Time
Wednesday, February 22, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Avram SidiTecnion-IIT, Haifa, Israel
We discuss some recent generalizations of Euler--Maclaurin expansions for the trapezoidal rule and of analogous asymptotic expansions for Gauss--Legendre quadrature, in the presence of arbitrary algebraic-logarithmic endpoint singularities. In addition of being of interest by themselves, these asymptotic expansions enable us to design appropriate variable transformations to improve the accuracies of these quadrature formulas arbitrarily. In general, these transformations are singular, and their singularities can be adjusted easily to achieve this improvement. We illustrate this issue with a numerical example involving Gauss--Legendre quadrature. We also discuss some recent asymptotic expansions of the coefficients of Legendre polynomial expansions of functions over a finite interval, assuming that the functions may have arbitrary algebraic-logarithmic interior and endpoint suingularities. These asymptotic expansions can be used to make definitive statements on the convergence acceleration rates of extrapolation methods as these are applied to the Legendre polynomial expansions.

### Two Weight inequality for the Hilbert transform

Series
Analysis Seminar
Time
Wednesday, February 1, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael LaceyGeorgia Tech
We continue with the proof of a real variable characterization of the two weight inequality for the Hilbert transform, focusing on a function theory in relevant for weights which are not doubling.

### Two weight inequality for the Hilbert transform

Series
Analysis Seminar
Time
Wednesday, January 25, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael LaceyGeorgia Tech
The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces, operator theory, and spectral theory, and what would be most useful is a characterization in the simplest real-variable terms. We show that the $L^2$ to $L^2$ inequality holds if and only if two $L^2$ to weak-$L^2$ inequalities hold. This is a corollary to a characterization in terms of a two-weight Poisson inequality, and a pair of testing inequalities on bounded functions. Joint work with Eric Sawyer, Chun-Yun Shen, and Ignacio Uriate-Tuero.

### On the behavior at infinity of solutions to difference equations in Schroedinger form

Series
Analysis Seminar
Time
Wednesday, January 18, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lillian WongGeorgia Tech
We offer several perspectives on the behavior at infinity of solutions of discrete Schroedinger equations. First we study pairs of discrete Schroedinger equations whose potential functions differ by a quantity that can be considered small in a suitable sense as the index n \rightarrow \infty. With simple assumptions on the growth rate of the solutions of the original system, we show that the perturbed system has a fundamental set of solutions with the same behavior at infinity, employing a variation-of-constants scheme to produce a convergent iteration for the solutions of the second equation in terms of those of the original one. We use the relations between the solution sets to derive exponential dichotomy of solutions and elucidate the structure of transfer matrices. Later, we present a sharp discrete analogue of the Liouville-Green (WKB) transformation, making it possible to derive exponential behavior at infinity of a single difference equation, by explicitly constructing a comparison equation to which our perturbation results apply. In addition, we point out an exact relationship connecting the diagonal part of the Green matrix to the asymptotic behavior of solutions. With both of these tools it is possible to identify an Agmon metric, in terms of which, in some situations, any decreasing solution must decrease exponentially.This talk is based on joint work with Evans Harrell.

### Dynamics of the support of the equilibrium measure in a quartic field

Series
Analysis Seminar
Time
Wednesday, December 7, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andrei Martinez FinkelshteinUniversity of Almeria, Spain
The asymptotic analysis of orthogonal polynomials with respect to a varying weight has found many interesting applications in approximation theory, random matrix theory and other areas. It has also stimulated a further development of the logarithmic potential theory, since the equilibrium measure in an external field associated with these weights enters the leading term of the asymptotics and its support is typically the place where zeros accumulate and oscillations occur. In a rather broad class of problems the varying weight on the real line is given by powers of a function of the form exp(P(x)), where P is a polynomial. For P of degree 2 the associated orthogonal polynomials can be expressed in terms of (varying) Hermite polynomials. Surprisingly, the next case, when P is of degree 4, is not fully understood. We study the equilibrium measure in the external field generated by such a weight, discussing especially the possible transitions between different configurations of its support. This is a joint work with E.A. Rakhmanov and R. Orive.