Multiscale analysis in Sobolev spaces on bounded domains with zero boundary values (Q2841067)

The authors study multi-scale approximation of Sobolev functions on a bounded domain \(\Omega\) using compactly supported radial basis functions (RBFs). The RBFs are constructed from a kernel generating the Sobolev space \(H^\tau(\mathbb{R}^d)\), \(\Phi\). Given a set of test sites \(X_j \subset \Omega\), the approximation to a function \(f\) at level \(j\) is formed by creating the interpolating function to \(f\) at the test sites \(X_j\) using the functions \( \displaystyle \delta_j^{-d}\Phi\left(\frac{\cdot - x_k}{\delta_j}\right)\), \(x_k \in X_j\), where \(\delta_j\) is a sequence of positive numbers decreasing to \(0\). In this study, the data sites are restricted so that the support of the interpolant lies within \(\Omega\). \vskip .1 in The main theorem of the paper states that if \(\Omega\) is a Lipschitz domain with an interior cone condition, the approximation to a function \(f \in H^\tau\) will converge linearly in the \(L^2\) norm to \(f\), provided that the data sites satisfy certain nice properties. Somewhat more specifically, if the data sites are quasi-uniform, and if the mesh norm of \(X_j\) decreases at a suitably fast rate from \(X_j\) to \(X_{j+1}\), and the value of \(\delta_j\) is sufficiently larger than the mesh norm of \(X_j\) for each \(j\), then the approximation will converge. \vskip .1 in The proof includes a Bernstein-type inequality for interpolants formed in the method described above. The authors also include a result on convergence for functions from Sobolev spaces with less smoothness than \(H^\tau(\mathbb{R}^d)\). Numerical examples are also given. The authors show numerical evidence that convergence is much improved for functions that vanish on the boundary of \(\Omega\), which is suggested as an area for further study.