Refinement equations and Feller operators (Q645110)

The paper studies the existence of integrable solutions \(f:\mathbb{R}^m \to \mathbb{R}\) of the fixed-point equation \[ f(x) = \int_\Omega |\det(K(\omega))| f(K(\omega)x-L(\omega)) \, dP(\omega). \] Here, \(K: \Omega \to \mathbb{R}^{m \times m}, \, L : \Omega \to \mathbb{R}^m\) are measurable functions and \((\Omega,\mathcal{A},P)\) is a probability space. The above fixed point equation generalizes the scaling equations occurring in wavelet analysis, and the results in this paper have potential applications in higher-dimensional wavelet theory. The authors establish a connection between the existence of solutions of the fixed-point equation and the problem of deciding whether the attractor of a suitable Feller operator is continuous. As an application, they obtain both necessary and sufficient conditions for the existence of integrable fixed points.