A density theorem for matrix-valued radial basis functions
Let (1) \(\Phi_{\ell,k}(x)=\{-\Delta I+\nabla\nabla^T\} \psi_{\ell,k}(x)\) be the general form of a positive definite, divergence-free and matrix-valued radial basis function (RBF), where \(\psi_{\ell,k}\in c^{2k}(\mathbb R^s)\) is a Wendland function (RBF), where \(\psi_{\ell,k}\in C^{2k}(\mathbb R^s)\) is a Wendland function of compact support with \(k\geq 2\), \(\ell=\lfloor s/2\rfloor+k+1\) and \(\widehat\psi_{\ell,k}(\xi)>0\) for all \(\xi\in\mathbb R^s\) (see H. Wendland [Adv. Comput. Math. 4, 389--396 (1995; Zbl 0838.41014)]. Hereby, \(\lfloor x\rfloor\) denotes the floor function which is the integer \(k\) such that \(k-1\leq x<k\), and \(\widehat f\) denotes the Fourier transform of a function f, i.e. \(\widehat f(\xi)=\int_{\mathbb R^s}e^{-ix\cdot \xi} f(x)\,dx\). Further, \(-Delta I+\nabla\nabla^T\) is the matrix-valued differential operator consisting of the Laplacian operator \(\Delta\), the gradient \(\nabla\), and the \(s\)-dimensional identity matrix \(I\). A function \(f\) is divergence-free if \(\nabla\cdot f\equiv 0\). The author establishes a density theorem as follows: Theorem. Let \(\Phi_{\ell,k}\) be given as in (1) and define the spaces \[ \nu:=\left\{\sum^N_{i=1}\Phi_{\ell,k}(\cdot-x_j)c_j:x_j\in\mathbb R^s,\quad c_j:x_j\in\mathbb R^s,\quad c_j\in\mathbb C^s\text{ for }1\leq j\leq N,N\in\mathbb N\right\} \] and \[ {\mathcal L}:=\{f\in H^1(\mathbb R^s;\mathbb C^s)\text{ such that }\nabla\cdot f\equiv 0\text{ a.e. on }\mathbb R^s\}. \] Then \(\nu\) is dense in \(\mathcal L\), i.e. any divergence-free vector-valued function \(f:\mathbb R^s\to\mathbb C^s\) contained in the Sobolev space \(H^1(\mathbb R^s;\mathbb C^s)\) can be approximated arbitrarily well by a linear combination of divergence-free matrix-valued RBFs generated by a Wendland function of compact support.
scientific article; zbMATH DE number 2170141