329 research outputs found
Elliptic scaling functions as compactly supported multivariate analogs of the B-splines
In the paper, we present a family of multivariate compactly supported scaling
functions, which we call as elliptic scaling functions. The elliptic scaling
functions are the convolution of elliptic splines, which correspond to
homogeneous elliptic differential operators, with distributions. The elliptic
scaling functions satisfy refinement relations with real isotropic dilation
matrices. The elliptic scaling functions satisfy most of the properties of the
univariate cardinal B-splines: compact support, refinement relation, partition
of unity, total positivity, order of approximation, convolution relation, Riesz
basis formation (under a restriction on the mask), etc. The algebraic
polynomials contained in the span of integer shifts of any elliptic scaling
function belong to the null-space of a homogeneous elliptic differential
operator. Similarly to the properties of the B-splines under differentiation,
it is possible to define elliptic (not necessarily differential) operators such
that the elliptic scaling functions satisfy relations with these operators. In
particular, the elliptic scaling functions can be considered as a composition
of segments, where the function inside a segment, like a polynomial in the case
of the B-splines, vanishes under the action of the introduced operator.Comment: To appear in IJWMI
On the Construction of Wavelets and Multiwavelets for General Dilation Matrices
This thesis is concerned with the construction of (pre-)wavelets and (pre-)multiwavelets. In particular, we identify minimal requirements such that a construction is still possible. To this end, we weaken the assumptions made in the definition of the multiresolution analysis. Based on this generalized multiresolution analysis, we develop construction procedures for compactly supported (pre-)wavelets and for compactly supported (pre-)multiwavelets. These construction procedures involve general dilation matrices which allow us to reduce the number of mother wavelets to a minimum. To illustrate the theory developed in this work, we choose exponential box splines as generators for the generalized multiresolution analysis and construct compactly supported (pre-)wavelets and (pre-)multiwavelets
Wavelet Decompositions of Nonrefinable Shift Invariant Spaces
AbstractThe motivation for this work is a recently constructed family of generators of shift invariant spaces with certain optimal approximation properties, but which are not refinable in the classical sense. We try to see whether, once the classical refinability requirement is removed, it is still possible to construct meaningful wavelet decompositions of dilates of the shift invariant space that are well suited for applications
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