Accuracy of lattice translates of several multidimensional refinable functions (Q1268712)

Complex-valued functions \(f_1,\dots, f_r\) on \(\mathbb{R}^d\) are refinable if they are linear combinations of finitely many of the rescaled and translated functions \(f_i(Ax-k)\), 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 functions 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 \(f(x)= (f_1(x),\dots, f_r(x))^T\). The accuracy of \(f\) is the highest degree \(p\) such that all multivariate polynomials \(q\) with \(\text{degree}(q)< p\) are exactly reproduced from linear combinations of translates of \(f_1,\dots, f_r\) along the lattice \(\Gamma\). In this paper, we determine the accuracy \(p\) from the matrices \(c_k\). Moreover, we determine explicitly the coefficients \(y_{\alpha,i} (k)\) such that \(x^\alpha= \sum_{i=1}^r \sum_{k\in\Gamma} y_{\alpha,i}(k) f_i(x+k)\). These coefficients are multivariate polynomials \(y_{\alpha,i} (x)\) of degree \(|\alpha |\) evaluated at lattice points \(k\in \Gamma\). The results are illustrated for the quincunx matrix, i.e. the \(2\times 2\)-matrix which represents a dilation by a factor \(\sqrt{2}\) and a rotation by \(\pi/4\) in 2-space.