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Search: All articles in the CJM digital archive with keyword convolution

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1. CJM Online first

da Silva, Genival; Kerr, Matt; Pearlstein, Gregory
 Arithmetic of degenerating principal variations of Hodge structure: examples arising from mirror symmetry and middle convolution We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi-Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions $1\leq d\leq6$ arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (which is $G_{2}$) of the family of 6-folds, and the theory of boundary components of Mumford-Tate domains. Keywords:variation of Hodge structure, limiting mixed Hodge structure, Calabi-Yau variety, middle convolution, Mumford-Tate groupCategories:14D07, 14M17, 17B45, 20G99, 32M10, 32G20

2. CJM 2012 (vol 66 pp. 102)

Birth, Lidia; Glöckner, Helge
 Continuity of convolution of test functions on Lie groups For a Lie group $G$, we show that the map $C^\infty_c(G)\times C^\infty_c(G)\to C^\infty_c(G)$, $(\gamma,\eta)\mapsto \gamma*\eta$ taking a pair of test functions to their convolution is continuous if and only if $G$ is $\sigma$-compact. More generally, consider $r,s,t \in \mathbb{N}_0\cup\{\infty\}$ with $t\leq r+s$, locally convex spaces $E_1$, $E_2$ and a continuous bilinear map $b\colon E_1\times E_2\to F$ to a complete locally convex space $F$. Let $\beta\colon C^r_c(G,E_1)\times C^s_c(G,E_2)\to C^t_c(G,F)$, $(\gamma,\eta)\mapsto \gamma *_b\eta$ be the associated convolution map. The main result is a characterization of those $(G,r,s,t,b)$ for which $\beta$ is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed, as well as convolution of compactly supported $L^1$-functions and convolution of compactly supported Radon measures. Keywords:Lie group, locally compact group, smooth function, compact support, test function, second countability, countable basis, sigma-compactness, convolution, continuity, seminorm, product estimatesCategories:22E30, 46F05, 22D15, 42A85, 43A10, 43A15, 46A03, 46A13, 46E25

3. CJM 2011 (vol 64 pp. 1075)

Raja, Chandiraraj Robinson Edward
 A Stochastic Difference Equation with Stationary Noise on Groups We consider the stochastic difference equation $$\eta _k = \xi _k \phi (\eta _{k-1}), \quad k \in \mathbb Z$$ on a locally compact group $G$ where $\phi$ is an automorphism of $G$, $\xi _k$ are given $G$-valued random variables and $\eta _k$ are unknown $G$-valued random variables. This equation was considered by Tsirelson and Yor on one-dimensional torus. We consider the case when $\xi _k$ have a common law $\mu$ and prove that if $G$ is a distal group and $\phi$ is a distal automorphism of $G$ and if the equation has a solution, then extremal solutions of the equation are in one-one correspondence with points on the coset space $K\backslash G$ for some compact subgroup $K$ of $G$ such that $\mu$ is supported on $Kz= z\phi (K)$ for any $z$ in the support of $\mu$. We also provide a necessary and sufficient condition for the existence of solutions to the equation. Keywords:dissipating, distal automorphisms, probability measures, pointwise distal groups, shifted convolution powersCategories:60B15, 60G20

4. CJM 2010 (vol 63 pp. 222)

Wang, Jiun-Chau
 Limit Theorems for Additive Conditionally Free Convolution In this paper we determine the limiting distributional behavior for sums of infinitesimal conditionally free random variables. We show that the weak convergence of classical convolution and that of conditionally free convolution are equivalent for measures in an infinitesimal triangular array, where the measures may have unbounded support. Moreover, we use these limit theorems to study the conditionally free infinite divisibility. These results are obtained by complex analytic methods without reference to the combinatorics of c-free convolution. Keywords:additive c-free convolution, limit theorems, infinitesimal arraysCategories:46L53, 60F05

5. CJM 2004 (vol 56 pp. 871)

Schocker, Manfred
 Lie Elements and Knuth Relations A coplactic class in the symmetric group $\Sym_n$ consists of all permutations in $\Sym_n$ with a given Schensted $Q$-symbol, and may be described in terms of local relations introduced by Knuth. Any Lie element in the group algebra of $\Sym_n$ which is constant on coplactic classes is already constant on descent classes. As a consequence, the intersection of the Lie convolution algebra introduced by Patras and Reutenauer and the coplactic algebra introduced by Poirier and Reutenauer is the direct sum of all Solomon descent algebras. Keywords:symmetric group, descent set, coplactic relation, Hopf algebra,, convolution productCategories:17B01, 05E10, 20C30, 16W30

6. CJM 2001 (vol 53 pp. 565)

Hare, Kathryn E.; Sato, Enji
 Spaces of Lorentz Multipliers We study when the spaces of Lorentz multipliers from $L^{p,t} \rightarrow L^{p,s}$ are distinct. Our main interest is the case when $s Keywords:multipliers, convolution operators, Lorentz spaces, Lorentz-improving multipliersCategories:43A22, 42A45, 46E30 7. CJM 1999 (vol 51 pp. 96) Rösler, Margit; Voit, Michael  Partial Characters and Signed Quotient Hypergroups If$G$is a closed subgroup of a commutative hypergroup$K$, then the coset space$K/G$carries a quotient hypergroup structure. In this paper, we study related convolution structures on$K/G$coming from deformations of the quotient hypergroup structure by certain functions on$K$which we call partial characters with respect to$G$. They are usually not probability-preserving, but lead to so-called signed hypergroups on$K/G$. A first example is provided by the Laguerre convolution on$\left[ 0,\infty \right[$, which is interpreted as a signed quotient hypergroup convolution derived from the Heisenberg group. Moreover, signed hypergroups associated with the Gelfand pair$\bigl( U(n,1), U(n) \bigr)\$ are discussed. Keywords:quotient hypergroups, signed hypergroups, Laguerre convolution, Jacobi functionsCategories:43A62, 33C25, 43A20, 43A90
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