A construction of refinable sets for interpolating wavelets
Let \(\Phi\) be a finite set of mappings from \(X\) into \(X\). A subset \(V\) of \(X\) is said to be refinable if \(V\subseteq\cup_{\phi\in\Phi}\phi (V)\). This paper presents a general construction of refinable sets in \({\mathbb R}^d\) which admits a unique polynomial interpolation. Denote by \(\Pi^d_n\) the space of all polynomials in \(d\) variables of total degree less than or equal to \(n\). A function \(\phi\) is a squeezer if, for some \(\rho\in (0,1)\) and \(x_0\in{\mathbb R}^d, \phi(x)=\rho (x-x_0)+x_0, x\in{\mathbb R}^d.\) Given a set \(\Phi\) consisting of \(d+1\) suitable squeezers, the authors construct a refinable set \(V=\{x_{\alpha}: \alpha\in {\mathbb N}^d_0, | \alpha| \leq n\}\) satisfying that, for any \(\alpha\in {\mathbb N}^d_0\) with \(| \alpha| \leq n\), there is a polynomial \(p_{\alpha}\in \Pi^d_n\) satisfying \(p_{\alpha}(x_{\beta})=\delta_{\alpha,\beta}\) \(\forall x_{\beta}\in V\). It is important that such refinable sets may lead to a fast interpolation wavelet decomposition. Example concerning triangles is provided to illustrated the theory.
scientific article; zbMATH DE number 1267816