Smooth maps from clumpy data

We study an estimator for smoothing irregularly sampled data into a smooth map. The estimator has been widely used in astronomy, owing to its low level of noise; it involves a weight function - or smoothing kernel - w(theta ). We show that this estimator is not unbiased, in the sense that the expectation value of the smoothed map is not the underlying process convolved with w, but a convolution with a modified kernel weff(theta ). We show how to calculate weff for a given kernel w and investigate its properties. In particular, it is found that (1) weff is normalized, (2) has a shape ``similar'' to the original kernel w, (3) converges to w in the limit of high number density of data points, and (4) reduces to a top-hat filter in the limit of very small number density of data points. Hence, although the estimator is biased, the bias is well understood analytically, and since weff has all the desired properties of a smoothing kernel, the estimator is in fact very useful. We present explicit examples for several filter functions which are commonly used, and provide a series expression valid in the limit of a large density of data points.