8 Conclusions

In this paper we have studied from a statistical point of view the effect of smoothing irregularly sampled data. The main results obtained can be summarized in the following points.

1.

The mean smooth map, $\bigl\langle \tilde f(\vec\theta) \bigr\rangle$, is a convolution of the unknown field $f(\vec\theta)$ with an effective weight $w_{\rm eff}$;

2.

We have provided simple expressions to evaluate the effective weight. These expressions can be easily used, for example, to obtain numerical estimates of $w_{\rm eff}$;

3.

The effective weight $w_{\rm eff}(\vec\theta)$ and the weight function $w(\vec\theta)$ share the same support and have a similar "shape''. However, $w_{\rm eff}$ is broader than w, expecially for low densities of objects; moreover, $w_{\rm eff}$ has a discontinuity at the boundary of the support;

4.

We have shown that the density of objects $\rho$ (or $\rho/(1-P_0)$ for finite-support weight functions) is a natural upper limit for the effective weights;

5.

The weight number $\mathcal{N}$ has been shown to be the key factor that controls the convergence of $w_{\rm eff}$ to w. We have also shown that $\mathcal{N}_{\rm eff} > 1$;

6.

The effective weight converges to $w(\vec\theta)$ for large densities $\rho$, and to a top-hat function for low densities;

7.

We have provided an analytic expansion for $w_{\rm eff}$ which is shown to converge quickly to the exact weight function;

8.

Finally, we have considered three typical examples and shown the behavior of $w_{\rm eff}$ for different densities.

Given the wide use in astronomy of the smoothing technique considered in this paper, an exact statistical characterization of the expectation value of the smoothed map is probably interesting per se.

Finally, we notice that other methods different from Eqs. (5) or (3) can be used to obtain continuous maps from irregularly sampled data. In particular, triangulation techniques can represent an interesting alternative to the simple weighted average considered here (see, e.g., Bernardeau & van de Weygaert 1996; Schaap & van de Weygaert 2000).

Acknowledgements

We would like to thank the Referee for comments and suggestions that enabled us to improve this paper.