Smooth maps from clumpy data: Generalizations^*

Institut für Astrophysik und Extraterrestrische Forschung, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany

Corresponding author: M. Lombardi, lombardi@astro.uni-bonn.de

Received: 30 September 2002
Accepted: 28 May 2003

Abstract

In a series of papers [CITE] we studied in detail the statistical properties of an interpolation technique widely used in astronomy. In particular, we considered the average interpolated map and its covariance under the hypotheses that the map is obtained by smoothing unbiased measurements of an unknown field, and that the measurements are uniformly distributed on the sky. In this paper we generalize the results obtained to the case of observations carried out only on a finite field and distributed on the field with a non-uniform density. These generalizations, which are required in many astronomically relevant cases, still allow an exact, analytical solution of the problem. We also consider a number of properties of the interpolated map, and provide asymptotic expressions for the average map and the two-point correlation function which are valid at high densities.