1 Introduction

A common problem in astronomy is the smoothing of some irregularly sampled data into a continuous map. It is hard to list all possible cases where such a problem is encountered. To give a few examples, we mention the determination of the stellar distribution of our Galaxy, the mapping of column density of dark molecular clouds from the absorption of background stars, the determination of the distribution function in globular clusters from radial and, recently, proper motions of stars, the determination of cosmological large-scale structures from redshift surveys, and the mass reconstruction in galaxy clusters from the observed distortion of background galaxies using weak lensing techniques. Similarly, one-dimensional data, such as time series of some events, often need to be smoothed in order to obtain a real function.

The use of a smooth map is convenient for at least two reasons. First, a smooth map can be better analyzed than irregularly sampled data. Second, if the smoothing is done in a sufficiently coarse way, the smooth map is significantly less noisy than the individual data. The drawback to this last point is a loss in resolution, but often this is a price that we have to pay in order to obtain results with a decent signal-to-noise ratio.

In many cases the transformation of irregularly sampled data into a smooth map follows a standard approach. A positive weight function, describing the relative weight of a datum at the position $\vec \theta + \vec\phi$ on the point $\vec\theta$, is introduced. This function is generally of the form $w(\vec\phi)$, i.e. is independent of the absolute position $\vec\theta$ of the point, and actually often depends only on the separation $\vert \vec\phi \vert$. The weight function $w(\vec\phi)$ is also chosen so that its value is large when the datum is close to the point, i.e. when $\vert \vec\phi \vert$ is small, and vanishes when $\vert \vec\phi \vert$ is large. Then, the data are averaged using a weighted mean with the weights given by the function w. More precisely, calling $\hat f_n$ the nth datum obtained at the position $\vec\theta_n$, the smooth map is defined as

$$\tilde f(\vec\theta) = \frac{\sum_n \hat f_n w(\vec\theta - \vec\theta_n)}{\sum_n w(\vec\theta - \vec\theta_n)}\cdot$$ (1)

It is reasonable to assume that this standard approach works well and produces good results, and the frequent use of this estimator suggests this is the case. On the other hand, in our opinion the properties of the smoothing should be better characterized by means of rigorous calculations. Some authors have actually already studied the smoothing using approximations, and have obtained preliminary results (e.g., Lombardi & Bertin 1998; van Waerbeke 2000). To our knowledge, however, the general problem has not been fully addressed so far and in particular there are no exact results known.

In this paper we consider in detail the effect of the smoothing on irregularly sampled data and derive a number of exact properties for the resulting map. We assume that the measurements $\hat f_n$ are unbiased estimates of some unknown field $f(\vec\theta)$ at the positions $\vec\theta_n$, and we study the expectation value of the map $\tilde f(\vec\theta)$ using an ensemble average, i.e. taking the N positions $\bigl\{ \vec\theta_n \bigr\}$ as random variables. We then show that the expectation value for the smooth map of Eq. (1) is given by

$$\bigl\langle \tilde f(\vec\theta) \bigr\rangle = \int f(\ve... ...a') w_{\rm eff}(\vec\theta - \vec\theta') \, \rm d^2 \theta'.$$ (2)

Thus, $\bigl\langle \tilde f \bigr\rangle$ is the convolution of the unknown field f with an effective weight $w_{\rm eff}$which, in general, differs from the weight function w. We also show that $w_{\rm eff}$ has a "similar'' shape as w and converges to w for a large number of objects N, but in general $w_{\rm eff}$is broader than w. Moreover, the effective weight is normalized, so that no signal is "lost'' or "created.'' Finally, we obtain some analytical expansions for $w_{\rm eff}$, and investigate its behavior in a number of interesting cases.

A common alternative to Eq. (1) is a non-normalized weighted sum, defined as

$$\tilde f(\vec\theta) = \frac{1}{\rho} \sum_{n=1}^N \hat f_n w(\vec\theta - \vec\theta_n),$$ (3)

where w has been assumed to have unit integral [see below Eq. (6)]. The statistical properties of this estimator can be easily derived, and in particular we obtain for its expectation value [cf. Eq. (3)]

$$\bigl\langle \tilde f(\vec\theta) \bigr\rangle = \int f(\vec\theta') w(\vec\theta - \vec\theta') \, \rm d^2 \theta' .$$ (4)

However, we note that this non-normalized estimator is expected to be more noisy than the one defined in Eq. (1) because of sampling noise. For example, even in the case of a flat field $f(\vec\theta) = 1$ measured without errors (so that $\hat f_n = 1$) we expect to have a noisy map if we use Eq. (3). For this reason, whenever possible the estimator (1) should be used instead.

The paper is organized as follows. In Sect. 2 we derive some preliminary expressions for the mean value of the map of Eq. (1). These results are generalized to a variable number N of objects in Sect. 3. If the weight function $w(\vec\theta)$ is allowed to vanish, then some peculiarities arises. This case is considered in detail in Sect. 4. Section 5 is dedicated to general properties for the average of $\tilde f$ and related functions. In Sect. 6 we take an alternative approach which can be used to obtain an analytic expansion for $\tilde f$. In Sect. 7, we consider three specific examples of weight functions often used in practice. Finally, a summary of the results obtained in this paper is given in Sect. 8. A variation of the smoothing technique considered in this paper is briefly discussed in Appendix A.