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   -> Volume 7, Issue 2

Preprint: Wavelets on Manifolds
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Angela Kunoth (

PostPosted: Mon Jan 26, 1998 1:31 pm    
Subject: Preprint: Wavelets on Manifolds
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#4 Preprint: Wavelets on Manifolds

New Preprints by Dahmen and Schneider:

Wavelets on Manifolds I: Construction and Domain Decomposition

Wolfgang Dahmen and Reinhold Schneider

January 1998


The potential of wavelets as a discretization tool for the numerical
treatment of operator equations hinges on the validity of norm
equivalences for Besov or Sobolev spaces in terms of weighted sequence
norms of wavelet expansion coefficients and on certain cancellation
properties. These features are crucial for the construction of optimal
preconditioners, for matrix compression based on sparse
representations of functions and operators as well as for the design
and analysis of adaptive solvers. So far the availability of such
bases is confined to very simple domain geometries. This paper is
concerned with concepts that aim at expanding the applicability of
wavelet schemes. The central issue is to construct wavelet bases with
the desired properties on manifolds which can be represented as the
disjoint union of smooth parametric images of the standard cube. The
approach is based on the characterization of function spaces over such
a manifold in terms of product spaces where each factor is a
corresponding local function space subject to certain boundary
conditions. Wavelet bases for each factor can be obtained as
parametric liftings from bases on the standard cube satisfying
appropriate boundary conditions. The use of such bases for the
discretization of operator equations leads in a natural way to a
conceptually new domain decomposition method. It is shown to exhibit
the same favorable convergence properties for a wide range of elliptic
operator equations covering, in particular, operators of nonpositive
order. In this paper we address all three issues, namely, the
characterization of function spaces which is intimately intertwined
with the construction of the wavelets, their relevance with regard to
matrix compression and preconditioning and the domain decomposition

Key Words:
Topological isomorphisms, Sobolev spaces on manifolds, norm
equivalences, complementary boundary conditions, biorthogonal wavelet
bases, domain
decomposition, boundary integral equations.


Wavelets with Complementary Boundary Conditions
- Function Spaces on the Cube

Wolfgang Dahmen and Reinhold Schneider

January 1998

Abstract: This paper is concerned with the construction of
biorthogonal wavelet bases on n-dimensional cubes which provide Riesz
bases for Sobolev and Besov spaces with homogeneous Dirichlet boundary
conditions on part of the boundary. The essential point is that the
primal and dual wavelets satisfy certain corresponding complementary
boundary conditions. These results form the key ingredients of the
construction of wavelet bases on manifolds (see preprint mentioned
above) that have been developed for the treatment of operator
equations of positive and negative order.

Key Words:
Topological isomorphisms, Sobolev and Besov spaces,
biorthogonal wavelet bases, moment conditions, complementary
boundary conditions.
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