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Search: MSC category 44 ( Integral transforms, operational calculus )

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1. CMB 2012 (vol 57 pp. 289)

Ghasemi, Mehdi; Marshall, Murray; Wagner, Sven
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)

Nhan, Nguyen Du Vi; Duc, Dinh Thanh
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)

Oberlin, Daniel M.
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)

Iakovlev, Serguei
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)

He, Jianxun
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)

Marshall, M.
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)

Iakovlev, Serguei
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)

Marshall, M.
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)

Oberlin, Daniel M.
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)

Marshall, Murray A.
A Real Holomorphy Ring without the Schmüdgen 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)

Daly, James E.; Phillips, Keith L.
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

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