The construction of wavelets adapted to compact domains
Let \(E\) be a compact set in \(\mathbb R^d\) and \(\rho\) be a measure defined on \(E\). Under some assumptions on the structure of \(E\) and \(\rho\), the author provides a methodology to construct wavelets to the space \(L^2(E, \rho)\). The methodology includes four steps: defining the measure \(\rho\) on \(E\); finding a map from \(E\) onto the unit in \(\mathbb R^d\); constructing the maps along with a refinable vector field; and constructing the initial wavelet space. Examples of the construction for a rectangular surface, a triangular surface, and a smooth simplex in \(\mathbb R^{3}\) are given. An extension of the theory to triangulated manifolds of finite distortion in \(n\) dimensions is also explained.
scientific article; zbMATH DE number 6160928