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1. CJM 2005 (vol 57 pp. 598)
Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group Differential operators $D_x$, $D_y$, and $D_z$ are formed using the
action of the $3$-dimensional discrete Heisenberg group $G$ on a set
$S$, and the operators will act on functions on $S$. The Laplacian
operator $L=D_x^2 + D_y^2 + D_z^2$ is a difference operator with
variable differences which can be associated to a unitary
representation of $G$ on the Hilbert space $L^2(S)$. Using techniques
from harmonic analysis and representation theory, we show that the
Laplacian operator is locally solvable.
Keywords:discrete Heisenberg group,, unitary representation,, local solvability,, difference operator Categories:43A85, 22D10, 39A70 |
2. CJM 2003 (vol 55 pp. 401)
Gaussian Estimates in Lipschitz Domains We give upper and lower Gaussian estimates for the diffusion kernel of a
divergence and nondivergence form elliptic operator in a Lipschitz domain.
On donne des estimations Gaussiennes pour le noyau d'une diffusion,
r\'eversible ou pas, dans un domaine Lipschitzien.
Categories:39A70, 35-02, 65M06 |
3. CJM 2001 (vol 53 pp. 1057)
Potential Theory in Lipschitz Domains We prove comparison theorems for the probability of life in a
Lipschitz domain between Brownian motion and random walks.
On donne des th\'eor\`emes de comparaison pour la probabilit\'e de
vie dans un domain Lipschitzien entre le Brownien et de marches
al\'eatoires.
Categories:39A70, 35-02, 65M06 |
4. CJM 1997 (vol 49 pp. 944)
Approximation by multiple refinable functions We consider the shift-invariant space,
$\bbbs(\Phi)$, generated by a set $\Phi=\{\phi_1,\ldots,\phi_r\}$
of compactly supported distributions on $\RR$ when the vector of
distributions $\phi:=(\phi_1,\ldots,\phi_r)^T$ satisfies a system
of refinement equations expressed in matrix form as
$$
\phi=\sum_{\alpha\in\ZZ}a(\alpha)\phi(2\,\cdot - \,\alpha)
$$
where $a$ is a finitely supported sequence of $r\times r$ matrices
of complex numbers. Such {\it multiple refinable functions} occur
naturally in the study of multiple wavelets.
The purpose of the present paper is to characterize the {\it accuracy}
of $\Phi$, the order of the polynomial space contained in
$\bbbs(\Phi)$, strictly in terms of the refinement mask $a$. The
accuracy determines the $L_p$-approximation order of $\bbbs(\Phi)$ when
the functions in $\Phi$ belong to $L_p(\RR)$ (see Jia~[10]).
The characterization is achieved in terms of the eigenvalues and
eigenvectors of the subdivision operator associated with the mask $a$.
In particular, they extend and improve the results of Heil, Strang
and Strela~[7], and of Plonka~[16]. In addition, a
counterexample is given to the statement of Strang and Strela~[20]
that the eigenvalues of the subdivision operator determine the
accuracy. The results do not require the linear independence of
the shifts of $\phi$.
Keywords:Refinement equations, refinable functions, approximation, order, accuracy, shift-invariant spaces, subdivision Categories:39B12, 41A25, 65F15 |