A compactly supported scaling function can come from a refinement equation with infinitely many nonzero coefficients (an infinite mask). In this case we prove that the symbol of the mask must have the special rational form ã(Z) = b̃(Z)c̃(Z)/b̃(Z). Any finite combination of the shifts of a refinable function will have such a mask, and will be refinable. We also study compactly supported solutions o...

In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ and φ̃ in L2(R) satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψjk := 2j/2ψ(2j · − k) (j, k ∈ Z) form a Riesz basis for L2(R). If, in addition, φ lies in the Sobolev space H(R), t...

Cartoon-like images, i.e., C functions which are smooth apart from a C discontinuity curve, have by now become a standard model for measuring sparse (non-linear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit (almost) optimally sparse approximations within this model. However, all those results ar...

This paper presents a construction of compactly supported dual functions of a given box spline in L2(IR ). In particular, a concrete method for the construction of compactly supported dual functions of bivariate box splines of increasing smoothness is provided. Key-Words:multivariate biorthogonal wavelets, multivariate wavelets, box splines, matrix extension

Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise: to two dual Riesz bases of compactly supported wavelets. We give necessary ...

Let the manifold X parametrise a family of compact complex submanifolds of the complex (or CR) manifold Z. Under mild conditions the Penrose transform typically provides isomorphisms between a cohomology group of a holomorphic vector bundle V → Z and the kernel of a differential operator between sections of vector bundles over X. When the spaces in question are homogeneous for a group G the Pen...

In this paper we present an example of a refinement equation such that up to a multiplicative constant it has a unique compactly supported distribution solution while it can simultaneously have a compactly supported componentwise constant function solution that is not locally integrable. This leads to the conclusion that in general the componentwise polynomial solution cannot be globally identi...

For a given continuous compactly supported reenable function that is fundamental (that means, () =), this paper presents several methods to construct, directly from , compactly supported fundamental reenable functions with higher regularity. Asymptotic regularity analyses of the constructions are given. These constructions immediately provide multivariate interpolatory subdivision schemes that ...

Let M ¢ 2 be a fixed positive integer. A family of closed subspaces Vj , j √ Z , of L 2 , the space of all square integrable functions on the real line, is said to be a multiresolution of L 2 if the following conditions hold: ( i) Vj , Vj/1 , and f √ Vj if and only if f ( Mr) √ Vj/1 for all j √ Z ; ( ii ) < j√Z Vj is dense in L 2 and > j√Z Vj Å M; ( iii ) there exists a function f in V0 such th...