Expand all Collapse all | Results 1 - 6 of 6 |
1. CJM 2008 (vol 60 pp. 1283)
Remarks on Littlewood--Paley Analysis Littlewood--Paley analysis is generalized in
this article. We show that the compactness of the Fourier support
imposed on the analyzing function can be removed. We also prove
that the Littlewood--Paley decomposition of tempered distributions
converges under a topology stronger than the weak-star topology,
namely, the inductive limit topology. Finally, we construct a
multiparameter Littlewood--Paley analysis and obtain the
corresponding ``renormalization'' for the convergence of this
multiparameter Littlewood--Paley analysis.
Keywords:Littlewood--Paley analysis, distributions Category:42B25 |
2. CJM 2006 (vol 58 pp. 897)
Distributions invariantes sur les groupes rÃ©ductifs quasi-dÃ©ployÃ©s Soit $F$ un corps local non archim\'edien, et $G$ le groupe des
$F$-points d'un groupe r\'eductif connexe quasi-d\'eploy\'e d\'efini sur $F$.
Dans cet article, on s'int\'eresse aux distributions sur $G$ invariantes
par conjugaison, et \`a l'espace de leurs restrictions \`a l'alg\`ebre de
Hecke $\mathcal{H}$ des fonctions sur $G$ \`a support compact
biinvariantes par un sous-groupe d'Iwahori $I$ donn\'e. On montre tout
d'abord que les valeurs d'une telle distribution sur $\mathcal{H}$
sont enti\`erement d\'etermin\'ees par sa restriction au sous-espace de
dimension finie des \'el\'ements de $\mathcal{H}$ \`a support dans la
r\'eunion des sous-groupes parahoriques de $G$ contenant $I$. On utilise
ensuite cette propri\'et\'e pour montrer, moyennant certaines conditions
sur $G$, que cet espace est engendr\'e d'une part par certaines
int\'egrales orbitales semi-simples, d'autre part par les int\'egrales
orbitales unipotentes, en montrant tout d'abord des r\'esultats
analogues sur les groupes finis.
Keywords:reductive $p$-adic groups, orbital integrals, invariant distributions Categories:22E35, 20G40 |
3. CJM 2006 (vol 58 pp. 3)
The Functional Equation of Zeta Distributions Associated With Non-Euclidean Jordan Algebras This paper is devoted to the study of certain zeta distributions
associated with simple non-Euclidean Jordan algebras. An explicit
form of the corresponding functional equation and Bernstein-type
identities is obtained.
Keywords:Zeta distributions, functional equations, Bernstein polynomials, non-Euclidean Jordan algebras Categories:11M41, 17C20, 11S90 |
4. CJM 2003 (vol 55 pp. 292)
Infinitely Divisible Laws Associated with Hyperbolic Functions The infinitely divisible distributions on $\mathbb{R}^+$ of random
variables $C_t$, $S_t$ and $T_t$ with Laplace transforms
$$
\left( \frac{1}{\cosh \sqrt{2\lambda}} \right)^t, \quad \left(
\frac{\sqrt{2\lambda}}{\sinh \sqrt{2\lambda}} \right)^t, \quad \text{and}
\quad \left( \frac{\tanh \sqrt{2\lambda}}{\sqrt{2\lambda}} \right)^t
$$
respectively are characterized for various $t>0$ in a number of
different ways: by simple relations between their moments and
cumulants, by corresponding relations between the distributions and
their L\'evy measures, by recursions for their Mellin transforms, and
by differential equations satisfied by their Laplace transforms. Some
of these results are interpreted probabilistically via known
appearances of these distributions for $t=1$ or $2$ in the description
of the laws of various functionals of Brownian motion and Bessel
processes, such as the heights and lengths of excursions of a
one-dimensional Brownian motion. The distributions of $C_1$ and $S_2$
are also known to appear in the Mellin representations of two
important functions in analytic number theory, the Riemann zeta
function and the Dirichlet $L$-function associated with the quadratic
character modulo~4. Related families of infinitely divisible laws,
including the gamma, logistic and generalized hyperbolic secant
distributions, are derived from $S_t$ and $C_t$ by operations such as
Brownian subordination, exponential tilting, and weak limits, and
characterized in various ways.
Keywords:Riemann zeta function, Mellin transform, characterization of distributions, Brownian motion, Bessel process, LÃ©vy process, gamma process, Meixner process Categories:11M06, 60J65, 60E07 |
5. CJM 2002 (vol 54 pp. 945)
Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications |
Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications Given a homogeneous elliptic partial differential operator $L$ with constant
complex coefficients and a class of functions (jet-distributions) which
are defined on a (relatively) closed subset of a domain $\Omega$ in $\mathbf{R}^n$ and
which belong locally to a Banach space $V$, we consider the problem of
approximating in the norm of $V$ the functions in this class by ``analytic''
and ``meromorphic'' solutions of the equation $Lu=0$. We establish new Roth,
Arakelyan (including tangential) and Carleman type theorems for a large class
of Banach spaces $V$ and operators $L$. Important applications to boundary
value problems of solutions of homogeneous elliptic partial differential
equations are obtained, including the solution of a generalized Dirichlet
problem.
Keywords:approximation on closed sets, elliptic operator, strongly elliptic operator, $L$-meromorphic and $L$-analytic functions, localization operator, Banach space of distributions, Dirichlet problem Categories:30D40, 30E10, 31B35, 35Jxx, 35J67, 41A30 |
6. CJM 1997 (vol 49 pp. 3)
Sweeping out properties of operator sequences Let $L_p=L_p(X,\mu)$, $1\leq p\leq\infty$, be the usual Banach
Spaces of real valued functions on a complete non-atomic
probability space. Let $(T_1,\ldots,T_{K})$ be
$L_2$-contractions. Let $0<\varepsilon < \delta\leq1$. Call a
function $f$ a $\delta$-spanning function if $\|f\|_2 = 1$ and if
$\|T_kf-Q_{k-1}T_kf\|_2\geq\delta$ for each $k=1,\ldots,K$, where
$Q_0=0$ and $Q_k$ is the orthogonal projection on the subspace spanned
by $(T_1f,\ldots,T_kf)$. Call a function $h$ a
$(\delta,\varepsilon)$-sweeping function if $\|h\|_\infty\leq1$,
$\|h\|_1<\varepsilon$, and if
$\max_{1\leq k\leq K}|T_kh|>\delta-\varepsilon$ on a set of
measure greater than $1-\varepsilon$. The following is the main
technical result, which is obtained by elementary estimates. There
is an integer $K=K(\varepsilon,\delta)\geq1$ such that if $f$ is a
$\delta$-spanning function, and if the joint distribution
of $(f,T_1f,\ldots,T_Kf)$ is normal, then $h=\bigl((f\wedge
M)\vee(-M)\bigr)/M$
is a $(\delta,\varepsilon)$-sweeping function, for some $M>0$.
Furthermore, if $T_k$s are the averages of operators induced by
the iterates of a measure preserving ergodic transformation, then a
similar result is true without requiring that the joint distribution
is normal. This gives the following theorem on a sequence $(T_i)$ of
these averages. Assume that for each $K\geq1$ there is a subsequence
$(T_{i_1},\ldots,T_{i_K})$ of length $K$, and a $\delta$-spanning
function $f_K$ for this subsequence. Then for each $\varepsilon>0$
there is a function $h$,
$0\leq h\leq1$,
$\|h\|_1<\varepsilon$, such that $\limsup_iT_ih\geq\delta$ a.e..
Another application of the main result gives a refinement of a part
of Bourgain's ``Entropy Theorem'', resulting in a
different, self contained proof of that theorem.
Keywords:Strong and $\delta$-sweeping out, Gaussian distributions, Bourgain's entropy theorem. Categories:28D99, 60F99 |