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Accuracy of several multidimensional refinable distributions. (English) Zbl 0960.42016

Summary: Compactly supported distributions \(f_1,\dots, f_r\) on \(\mathbb{R}^d\) are refinable if each \(f_i\) is a finite linear combination of the rescaled and translated distributions \(f_j(Ax- k)\), where the translates \(k\) are taken along a lattice \(\Gamma\subset \mathbb{R}^d\) and \(A\) is a dilation matrix that expansively maps \(\Gamma\) into itself. Refinable distributions satisfy a refinement equation \(f(x)= \sum_{k\in\Lambda} c_kf(Ax- k)\), where \(\Lambda\) is a finite subset of \(\Gamma\), the \(c_k\) are \(r\times r\) matrices, and \(f= (f_1,\dots, f_r)^T\). The accuracy of \(f\) is the highest degree \(p\) such that all multivariate polynomials \(q\) with degree\((q)< p\) are exactly reproduced from linear combinations of translates of \(f_1,\dots, f_r\) along the lattice \(\Gamma\). We determine the accuracy \(p\) from the matrices \(c_k\). Moreover, we determine explicitly the coefficients \(y_{\alpha, i}(k)\) such that \(x^\alpha= \sum^r_{i= 1} \sum_{k\in\Gamma} y_{\alpha, i}(k) f_i(x+ k)\). These coefficients are multivariate polynomials \(y_{\alpha, i}(x)\) of degree \(|\alpha|\) evaluated at lattice points \(k\in\Gamma\).

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A25 Rate of convergence, degree of approximation
39B62 Functional inequalities, including subadditivity, convexity, etc.
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