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- W2897070083 abstract "Let $Gamma$ be a crystal group in $mathbb R^d$. A function $varphi:mathbb R^dlongrightarrow mathbb C$ is said to be {em crystal-refinable} (or $Gamma-$refinable) if it is a linear combination of finitely many of the rescaled and translated functions $varphi(gamma^{-1}(ax))$, where the {em translations} $gamma$ are taken on a crystal group $Gamma$, and $a$ is an expansive dilation matrix such that $aGamma a^{-1}subsetGamma.$ A $Gamma-$refinable function $varphi: mathbb R^d rightarrow mathbb C$ satisfies a refinement equation $varphi(x)=sum_{gammainGamma}d_gamma varphi(gamma^{-1}(ax))$ with $d_gamma in mathbb C$. Let $mathcal S(varphi)$ be the linear span of ${varphi(gamma^{-1}(x)): gamma in Gamma}$ and $mathcal{S}^h={f(x/h):finmathcal{S(varphi)}}$. One important property of $mathcal S(varphi)$ is, how well it approximates functions in $L^2(mathbb R^d)$. This property is very closely related to the {em crystal-accuracy} of $mathcal S(varphi)$, which is the highest degree $p$ such that all multivariate polynomials $q(x)$ of ${rm degree}(q)<p$ are exactly reproduced from elements in $mathcal S(varphi)$. In this paper, we determine the accuracy $p$ from the coefficients $d_gamma$. Moreover, we obtain from our conditions, a characterization of accuracy for a particular lattice refinable vector function $F$, which simplifies the classical conditions." @default.
- W2897070083 created "2018-10-26" @default.
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- W2897070083 date "2017-01-28" @default.
- W2897070083 modified "2023-09-27" @default.
- W2897070083 title "Approximation by crystal-refinable function" @default.
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