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:
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.
Wednesday, December 7, 2011 - 14:00 , Location: Skiles 006 , Andrei Martinez Finkelshtein , University of Almeria, Spain , email@example.com , Organizer: Jeff Geronimo
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.
Thursday, December 1, 2011 - 14:00 , Location: Skiles 006 , Pierre Moussa , CEA/Saclay, Service de Physique Theorique, France , Organizer: Jeff Geronimo
The term "BMV Conjecture" was introduced in 2004 by Lieb and Seiringer for a conjecture introduced in 1975 by Bessis, Moussa and Villani, and they also introduced a new form for it : all coefficients of the polynomial Tr(A+xB)^k are non negative as soon as the hermitian matrices A and B are positive definite. A recent proof of the conjecture has been given recently by Herbert Stahl. The question occurs in various domains: complex analysis, combinatorics, operator algebras and statistical mechanics.
Weierstrass Theorem for homogeneous polynomials on convex bodies and rate of approximation of convex bodies by convex algebraic level surfacesTuesday, November 29, 2011 - 14:00 , Location: Skiles 006 , Prof. Andras Kroo , Hungarian Academy of Sciences , Organizer: Doron Lubinsky
By the classical Weierstrass theorem, any function continuous on a compact set can be uniformly approximated by algebraic polynomials. In this talk we shall discuss possible extensions of this basic result of analysis to approximation by homogeneous algebraic polynomials on central symmetric convex bodies. We shall also consider a related question of approximating convex bodies by convex algebraic level surfaces. It has been known for some time time that any convex body can be approximated arbitrarily well by convex algebraic level surfaces. We shall present in this talk some new results specifying rate of convergence.
Wednesday, November 9, 2011 - 14:00 , Location: Skiles 006 , Aleks Ignjatovic , University of New South Wales , Organizer: Doron Lubinsky
Chromatic derivatives are special, numerically robust linear differential operators which provide a unification framework for a broad class of orthogonal polynomials with a broad class of special functions. They are used to define chromatic expansions which generalize the Neumann series of Bessel functions. Such expansions are motivated by signal processing; they grew out of a design of a switch mode power amplifier. Chromatic expansions provide local signal representation complementary to the global signal representation given by the Shannon sampling expansion. Unlike the Taylor expansion which they are intended to replace, they share all the properties of the Shannon expansion which are crucial for signal processing. Besides being a promising new tool for signal processing, chromatic derivatives and expansions have intriguing mathematical properties connecting in a novel way orthogonal polynomials with some familiar concepts and theorems of harmonic analysis. For example, they introduce novel spaces of almost periodic functions which naturally correspond to a broad class of families of orthogonal polynomials containing most classical families. We also present a conjecture which generalizes the Paley Wiener Theorem and which relates the growth rate of entire functions with the asymptotic behavior of the recursion coefficients of a corresponding family of orthogonal polynomials.