Expand all Collapse all | Results 1 - 11 of 11 |
1. CMB 2012 (vol 57 pp. 289)
Closure of the Cone of Sums of $2d$-powers in Certain Weighted $\ell_1$-seminorm Topologies In a paper from 1976, Berg, Christensen and Ressel prove that the
closure of the cone of sums of squares $\sum
\mathbb{R}[\underline{X}]^2$ in the polynomial ring
$\mathbb{R}[\underline{X}] := \mathbb{R}[X_1,\dots,X_n]$ in the
topology induced by the $\ell_1$-norm is equal to
$\operatorname{Pos}([-1,1]^n)$, the cone consisting of all polynomials
which are non-negative on the hypercube $[-1,1]^n$. The result is
deduced as a corollary of a general result, established in the same
paper, which is valid for any commutative semigroup.
In later work, Berg and Maserick and Berg, Christensen and Ressel
establish an even more general result, for a commutative semigroup
with involution, for the closure of the cone of sums of squares of
symmetric elements in the weighted $\ell_1$-seminorm topology
associated to an absolute value.
In the present paper we give a new proof of these results which is
based on Jacobi's representation theorem from 2001. At the same time,
we use Jacobi's representation theorem to extend these results from
sums of squares to sums of $2d$-powers, proving, in particular, that
for any integer $d\ge 1$, the closure of the cone of sums of
$2d$-powers $\sum \mathbb{R}[\underline{X}]^{2d}$ in
$\mathbb{R}[\underline{X}]$ in the topology induced by the
$\ell_1$-norm is equal to $\operatorname{Pos}([-1,1]^n)$.
Keywords:positive definite, moments, sums of squares, involutive semigroups Categories:43A35, 44A60, 13J25 |
2. CMB 2011 (vol 55 pp. 355)
Convolution Inequalities in $l_p$ Weighted Spaces Various weighted $l_p$-norm inequalities in convolutions are derived
by a simple and general principle whose $l_2$ version was obtained by
using the theory of reproducing kernels. Applications to the Riemann zeta
function and a difference equation are also considered.
Keywords:inequalities for sums, convolution Categories:26D15, 44A35 |
3. CMB 2011 (vol 55 pp. 815)
Restricted Radon Transforms and Projections of Planar Sets We establish a mixed norm estimate for the Radon transform in
$\mathbb{R}^2$ when the set of directions has fractional dimension.
This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to
a conjecture analogous to a well-known conjecture of Furstenberg.
Categories:44A12, 28A78 |
4. CMB 2007 (vol 50 pp. 547)
Inverse Laplace Transforms Encountered in Hyperbolic Problems of Non-Stationary Fluid-Structure Interaction |
Inverse Laplace Transforms Encountered in Hyperbolic Problems of Non-Stationary Fluid-Structure Interaction The paper offers a study of the inverse Laplace
transforms of the functions $I_n(rs)\{sI_n^{'}(s)\}^{-1}$ where
$I_n$ is the modified Bessel function of the first kind and $r$ is
a parameter. The present study is a continuation of the author's
previous work %[\textit{Canadian Mathematical Bulletin} 45]
on the
singular behavior of the special case of the functions in
question, $r$=1. The general case of $r \in [0,1]$ is addressed,
and it is shown that the inverse Laplace transforms for such $r$
exhibit significantly more complex behavior than their
predecessors, even though they still only have two different types
of points of discontinuity: singularities and finite
discontinuities. The functions studied originate from
non-stationary fluid-structure interaction, and as such are of
interest to researchers working in the area.
Categories:44A10, 44A20, 33C10, 40A30, 74F10, 76Q05 |
5. CMB 2004 (vol 47 pp. 389)
An Inversion Formula of the Radon Transform Transform on the Heisenberg Group In this paper we give an inversion formula of the Radon transform on the
Heisenberg group by using the wavelets defined in [3]. In addition, we
characterize a space such that the inversion formula of the Radon transform
holds in the weak sense.
Keywords:wavelet transform, Radon transform, Heisenberg group Categories:43A85, 44A15 |
6. CMB 2003 (vol 46 pp. 400)
Approximating Positive Polynomials Using Sums of Squares The paper considers the relationship between positive polynomials,
sums of squares and the multi-dimensional moment problem in the
general context of basic closed semi-algebraic sets in real $n$-space.
The emphasis is on the non-compact case and on quadratic module
representations as opposed to quadratic preordering presentations.
The paper clarifies the relationship between known results on the
algebraic side and on the functional-analytic side and extends these
results in a variety of ways.
Categories:14P10, 44A60 |
7. CMB 2002 (vol 45 pp. 399)
On the Singular Behavior of the Inverse Laplace Transforms of the Functions $\frac{I_n(s)}{s I_n^\prime(s)}$ |
On the Singular Behavior of the Inverse Laplace Transforms of the Functions $\frac{I_n(s)}{s I_n^\prime(s)}$ Exact analytical expressions for the inverse Laplace transforms of
the functions $\frac{I_n(s)}{s I_n^\prime(s)}$ are obtained in the
form of trigonometric series. The convergence of the series is
analyzed theoretically, and it is proven that those diverge on an
infinite denumerable set of points. Therefore it is shown that the
inverse transforms have an infinite number of singular points. This
result, to the best of the author's knowledge, is new, as the
inverse transforms of $\frac{I_n(s)}{s I_n^\prime(s)}$ have
previously been considered to be piecewise smooth and continuous.
It is also found that the inverse transforms have an infinite
number of points of finite discontinuity with different left- and
right-side limits. The points of singularity and points of finite
discontinuity alternate, and the sign of the infinity at the
singular points also alternates depending on the order $n$. The
behavior of the inverse transforms in the proximity of the singular
points and the points of finite discontinuity is addressed as well.
Categories:65R32, 44A10, 44A20, 74F10 |
8. CMB 2001 (vol 44 pp. 223)
Extending the Archimedean Positivstellensatz to the Non-Compact Case A generalization of Schm\"udgen's Positivstellensatz is given which holds
for any basic closed semialgebraic set in $\mathbb{R}^n$ (compact or not).
The proof is an extension of W\"ormann's proof.
Categories:12D15, 14P10, 44A60 |
9. CMB 2000 (vol 43 pp. 472)
An Estimate For a Restricted X-Ray Transform This paper contains a geometric proof of an estimate for a restricted
x-ray transform. The result complements one of A.~Greenleaf and
A.~Seeger.
Category:44A15 |
10. CMB 1999 (vol 42 pp. 354)
A Real Holomorphy Ring without the SchmÃ¼dgen Property A preordering $T$ is constructed in the polynomial ring $A = \R
[t_1,t_2, \dots]$ (countably many variables) with the following two
properties: (1)~~For each $f \in A$ there exists an integer $N$
such that $-N \le f(P) \le N$ holds for all $P \in \Sper_T(A)$.
(2)~~For all $f \in A$, if $N+f, N-f \in T$ for some integer $N$,
then $f \in \R$. This is in sharp contrast with the
Schm\"udgen-W\"ormann result that for any preordering $T$ in a
finitely generated $\R$-algebra $A$, if property~(1) holds, then
for any $f \in A$, $f > 0$ on $\Sper_T(A) \Rightarrow f \in T$.
Also, adjoining to $A$ the square roots of the generators of $T$
yields a larger ring $C$ with these same two properties but with
$\Sigma C^2$ (the set of sums of squares) as the preordering.
Categories:12D15, 14P10, 44A60 |
11. CMB 1998 (vol 41 pp. 392)
A note on $H^1$ multipliers for locally compact Vilenkin groups Kitada and then Onneweer and Quek have investigated multiplier
operators on Hardy spaces over locally compact Vilenkin groups. In
this note, we provide an improvement to their results for the Hardy
space $H^1$ and provide examples showing that our result applies to a
significantly larger group of multipliers.
Categories:43A70, 44A35 |