Accuracy of several multidimensional refinable distributions |
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Authors: | Carlos Cabrelli Chritopher Heil Ursula Molter |
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Affiliation: | (1) Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina;(2) CONICET, Rivadavia 1917, 1033 Buenos Aires, Argentina;(3) School of Mathematics, Georgia Institute of Technology, 30332-0160 Atlanta, Georgia, USA |
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Abstract: | Compactly supported distributions f1,..., fr on ℝd are fefinable if each fi is a finite linear combination of the rescaled and translated distributions fj(Ax−k), where the translates k are taken along a lattice Γ ⊂ ∝d and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x)=Σk∈Λ ck f(Ax−k), where Λ is a finite subset of Γ, the ck are r×r matrices, and f=(f1,...,fr)T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q) 1,...,fr along the lattice Γ. We determine the accuracy p from the matrices ck. Moreover, we determine explicitly the coefficients yα,i(k) such that xα=Σ i=1 r Σk∈Γyα,i(k) fi(x+k). These coefficients are multivariate polynomials yα,i(x) of degree |α| evaluated at lattice points k∈Γ. |
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Keywords: | Primary 41A25 secondary 39B62 65D15 |
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