Abstract:
This talk will center on properties of zeros of Bessel functions that
are related to various isoperimetric inequalities for the low
characteristic frequencies of homogeneous membranes and plates. These
correspond to optimal inequalities for the eigenvalues of the Laplacian
and biharmonic operators under suitable boundary conditions. For example,
by means of a simple identity we recover Lorch's result that
$j_{p,1}^2-(p+1)(p+5)$ increases from $0$ to infinity as $p$ runs from
$-1$ to infinity (here $j_{p,1}$ denotes the first positive zero of the
Bessel function $J_p(x)$). Several other inequalities for Bessel function
zeros which are suggested by variational theory or conjectured eigenvalue
inequalities will also be discussed.
Richard Askey (University of Wisconsin) "Bessel functions and how to use them when considering more general classes of functions"
Abstract:
From the time of Sturm's work on comparison theorems, one role Bessel
functions have played is to suggest results for more general classes of
functions. Some examples will be given, both of cases where more general
results have been found and ones where we are still looking to see how to
extend known results about Bessel functions.
James A. Donaldson (Howard University),
``The Linear Shallow Water Theory and Dirichlet-Neumann Operators''
Abstract
In obtaining a mathematical justification for the shallow water theory for
the case where the bottom of the region of the water is horizontal, the
Fourier
transformation plays a primary role; for the case where the bottom of the
region
of the water is non-horizontal, a generalized Fourier transformation
plays a
key role in the two-dimensional theory, and spectral theory of
self-adjoint
operators plays a similar role in the three-dimensional theory. We show
that
Dirichlet-Neumann operators appear naturally in the treatment of these
cases,
and discuss some of their properties.
This is joint work with Daniel A. Williams.
Jean-Pierre Kahane (Universite de Paris - Orsay)
Amram Meir (York University),
Degree distribution in random trees
Abstract:
A. McD. Mercer (University of Guelph), On
"deleting a row and column" from a differential operator.
Abstract:
Mark Pinsky (Northwestern University),
"Pointwise convergence of the Fourier integral and related orthogonal
expansions in several variables"
Abstract:
P. G. Rooney (University of Toronto), "On the Hankel transformation".
Abstract:
Cora Sadosky (Howard University), "Restricted BMO in product spaces".
Abstract:
Walter Van Assche,
Zeros of orthogonal polynomials and eigenvalues of matrices
Orthonormal polynomials on the real line always satisfy a
three-term recurrence relation
$$ x p_n(x) = a_{n+1}p_{n+1}(x) + b_n p_n(x) + a_n p_{n-1}(x), $$
where $a_n > 0$ and $b_n$ is real. This can be written as
$$ \left( \matrix{ b_0 & a_1 & & & \cr
a_1 & b_1 & a_2 & & \cr
& a_2 & b_2 & \ddots & \cr
& & \ddots & \ddots & \cr
& & & a_{n-1} & b_{n-1}}
\right)
\left( \matrix{p_0(x) \cr p_1(x) \cr p_2(x) \cr \vdots \cr p_{n-1}(x)}
\right) =
x \left( \matrix{p_0(x) \cr p_1(x) \cr p_2(x) \cr \vdots \cr p_{n-1}(x)}
\right) - a_n p_n(x)
\left( \matrix{0 \cr 0 \cr \vdots \cr 0 \cr 1} \right) , $$
which shows that a zero $x_{j,n}$ of $p_n$ is an eigenvalue of the
$n\times n$
Jacobi matrix containing the recurrence coefficients, with eigenvector
$\left( p_0(x_{j,n}) , p_1(x_{j,n}) , \cdots \, p_{n-1}(x_{j,n}) \right)$.
A similar connection exists between zeros of orthogonal polynomials on
the unit circle and certain Hessenberg matrices which are almost unitary.
Moreover this connection between zeros and eigenvalues can also be
extended to matrix valued polynomials.
We will show how one can obtain properties of zeros of orthogonal polynomials
by means of results from linear algebra. In particular interlacing properties,
Gauss quadrature (on the real line and on the unit citcle), and
monotonicity properties of the zeros as a function of the parameters.
By considering the infinite Jacobi matrix as an operator acting on $\ell_2$,
we will also show how results from operator theory give interesting
results for the corresponding orthogonal polynomials.
\bye
revised May 28, 1995
The ``Shallow Water'' Theory provides an example of the approximation of
the solution of an initial-boundary value problem for an elliptic partial
differential equation by the solution of an initial-value problem for a
hyperbolic partial differential equation. Formal and rigorous derivations
of the this theory are given and a mathematical justification for the
theory
is obtained.
"Summability, order and products of Dirichlet series".
Abstract:
This talk will be mainly historical, but also contains a few
new results on some questions left open by Stieltjes, Landau
and H.Bohr. In particular, results and problems on the
Lindeloef function for ordinary Dirichlet series will be
considered.
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\begin{center}
{\large
Degree distribution in random trees}
\\
\normalsize
\bigskip
Amram Meir\\
\bigskip
\end{center}
Let ${\cal F}$ be a simply generated family of random rooted trees
characterized by a functional relation $y = x\varphi(y)$, where $\varphi(t) =
\sum^\infty_0 c_k t^k$ satisfies certain analyticity conditions.
Let $N_m(T_n)$ denote the (weighted) number of nodes of out-degree $m$ in
$T_n \in {\cal F}$. It was established in an earlier paper by Meir and
Moon that
$$
E\{ N_m(T_n) : T_n \in {\cal F}\} \sim \frac{c_m\tau^m}{\varphi(\tau)} n ,
$$
where $\tau$ is the solution of $\tau\varphi'(\tau) = \varphi(\tau)$. V.T.
S\'os and I raised the question whether the proportion of nodes of
out-degree $m$
{\em on any level} $k$ is asymptotically the same as in the tree itself.
This is indeed the case for all $k = {\cal O}(n^{1/2})$ and $m = o(k)$.
This covers almost all of the relevant cases.
\end{document}
If we delete a row and its corresponding column from a
real symmetric matrix then the eigenvalues of the
resulting matrix interlace those of the original one.
There is an analogous result for a certain class of
differential operators. Furthermore, in this more
general context, large families of complete
orthogonal systems of functions make their appearance.
We review our recent work [AMS Notices, vol 42, March 1995, pp. 330-334.]
on the convergence of Fourier expansions at a pre-assigned point, for
piecewise smooth functions on Euclidean space. The neccessary and
sufficient condition for convergence of the spherical partial sums is
expressed in terms of the smoothness of the radial averages of the
function w.r.t. the pre-assigned point. Extensions to Jacobi polynomial
expansions and Hermite polynomial expansions in several variables are
formulated and proved.
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\begin{center}
{\large
On the Hankel transformation
}\\
\normalsize
\bigskip
P. G. Rooney\\
\bigskip
\end{center}
The Hankel transformation of order $\nu > -1$, $H_\nu$ is defined for
suitable functions $f$ by
$$(H_\nu f)(x) = \int_0^\infty (xt)^\frac12 J_\nu(xt)f(t)dt.$$
Its boundedness and range will be reviewed on the spaces ${\cal
L}_{\mu,p}$, defined for $\mu \in {\bf R},\; 1~<~p~<~\infty$, by the norm
$$ \parallel f \parallel_{\mu,p} \;= \left\{ \int_0^\infty |x^\mu f(x)|
\frac{dx}{x} \right\}^{\frac{1}{p}}.$$
It will be seen that the transforms of functions in suitable ${\cal
L}_{\mu,p}$ spaces satisfy Lipschitz and integral Lipschitz conditions.
This is joint work with P. Heywood.
\end{document}
The class BMOr, "restricted BMO," appears naturally in several
problems of harmonic analysis in product spaces, as those of
interpolation and approximation by analytic functions in the
polydisk, or of boundedness of the product Hilbert transform in
weighted Lebesgue spaces. First introduced to characterize the
symbols of bounded Hankel operators acting in the multidimensional
torus, BMOr provides necessary and sufficient conditions for
solutions to polydisk versions of Pick-Nevanlinna and Caratheodory-
Fejer problems. The characterization of weights in terms of
logarithmic BMOr provides new insights into function theory in
product spaces. This is joint work with Mischa Cotlar.
%Lee Lorch meeting, June 9 -10, 1995
% plain TeX
%
\magnification=1200
\centerline{\bf Zeros of orthogonal polynomials and eigenvalues of matrices}
\centerline{by Walter Van Assche\footnote{$^*$}{Senior Research Associate
of the Belgian National Fund for Scientific Research}}
\centerline{Katholieke Universiteit Leuven}
\bigskip