## Seminars and Colloquia by Series

Wednesday, April 25, 2012 - 15:30 , Location: Skiles 005 , Konstantin Oskolkov , University of South Carolina , Organizer: Michael Lacey
Wednesday, April 18, 2012 - 14:00 , Location: Skiles 005 , Kelly Bickel , Washington University - St. Louis , Organizer: Brett Wick
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.
Wednesday, April 11, 2012 - 14:00 , Location: Skiles 005 , Vladimir Eiderman , University of Wisconsin , Organizer: Michael Lacey
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.
Monday, March 26, 2012 - 14:00 , Location: Skiles 114 , Dan Timotin , Indiana University and Mathematical Institute of Romania , Organizer:
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.
Wednesday, March 14, 2012 - 14:00 , Location: Skiles 006 , Vygantas Paulauskas , Vilnius University , Organizer: Michael Lacey
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]
Wednesday, March 7, 2012 - 14:00 , Location: Skiles 005 , Mark Lewko , University of Texas , Organizer: Michael Lacey
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.
Wednesday, February 22, 2012 - 14:00 , Location: Skiles 006 , Prof. Avram Sidi , Tecnion-IIT, Haifa, Israel , Organizer: Doron Lubinsky
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.
Wednesday, February 1, 2012 - 14:00 , Location: Skiles 005 , Michael Lacey , Georgia Tech , Organizer: Michael Lacey
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.
Wednesday, January 25, 2012 - 14:00 , Location: Skiles 005 , Michael Lacey , Georgia Tech , Organizer: Michael Lacey
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.
Wednesday, January 18, 2012 - 14:00 , Location: Skiles 005 , Lillian Wong , Georgia Tech , Organizer: Brett Wick
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.