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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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

The problem of weighted polynomial approximation of continuousfunctionson the real line was posted by S. Bernstein in 1924. It asks for adescription of theset of weights such that polynomials are dense in the space of continuousfunctions withrespect to the corresponding weighted uniform norm. Throughout the 20thcentury Bernstein's problem was studied by many prominent analysts includingAhkiezer, Carleson, Mergelyan andM. Riesz.In my talk I will discuss some of the complex analytic methods that can beapplied in Bernstein's problem along with a recently found solution.

Series: Analysis Seminar

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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 arbirarily well by convex algebraic level surfaces. We
shall present in this talk some new results specifying rate of convergence.

Series: Analysis Seminar

Expansion in a wavelet basis provides useful information ona function in different positions and length-scales. The simplest example of wavelets are the Haar functions, which are just linearcombinations of characteristic functions of cubes, but often moresmoothness is preferred. It is well-known that the notion of Haarfunctions carries over to rather general abstract metric spaces. Whatabout more regular wavelets? It turns out that a neat construction canbe given, starting from averages of the indicator functions over arandom selection of the underlying cubes. This is yet anotherapplication of such probabilistic averaging methods in harmonicanalysis. The talk is based on joint work in progress with P. Auscher.

Series: Analysis Seminar

We consider ensembles of $N \times N$ Hermitian Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. Assuming sufficient regularity for the probability density function of the entries, we show that the expectation of the density of states on arbitrarily small intervals converges to the semicircle law, as $N$ tends to infinity.