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Series: Analysis Seminar

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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]

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.