2 Definitions and first results

Suppose one wants to measure an unknown field $f(\vec\theta)$, a function of the "position'' $\vec\theta$. [What $\vec\theta$ really means is totally irrelevant for our discussion. For example, $\vec\theta$ could represent the position of an object on the sky, the time of some observation, or the wavelength of a spectral feature. In the following, to focus on a specific case, we will assume that $\vec\theta$ represents a position on the sky and thus we will consider it as a two-dimensional variable.] Suppose also that we can obtain a total of N unbiased estimates $\hat f_n$ for fat some points $\bigl\{ \vec\theta_n \bigr\}$, and that each point can freely span a field $\Omega$ of surface A ($\Omega$ represents the area of the survey, i.e. the area where data are available). The points $\bigl\{ \vec\theta_n \bigr\}$, in other words, are taken to be independent random variables with a uniform probability distribution and density $\rho = N / A$ inside the set $\Omega$ of their possible values. We can then define the smooth map of Eq. (1), or more explicitly

$$\tilde f(\vec\theta) = \frac{\sum_{n=1}^N \hat f_n w(\vec\... ...vec\theta_n)}{\sum_{n=1}^N w(\vec\theta - \vec\theta_n)}\cdot$$ (5)

In the rest of this paper we study the expectation value $\bigl\langle \tilde f(\vec\theta) \bigr\rangle$ of $\tilde f(\vec\theta)$ (an alternative weighting scheme is briefly discussed in Appendix A). To simplify the notation we will assume, without loss of generality, that the weight function $w(\vec\theta)$ is normalized, i.e.

$$\int_\Omega w(\vec\theta) \, \rm d^2 \theta = 1 .$$ (6)

In order to obtain the ensemble average of $\tilde f$ we need to average over all possible measurements at each point, i.e. $\hat f_n$, and over all possible positions $\bigl\{ \vec\theta_n \bigr\}$for the N points. The first average is trivial, since $\tilde f(\vec\theta)$ is linear on the data $\hat f_n$ and the data are unbiased, so that $\bigl\langle \hat f_n \bigr\rangle = f(\vec\theta_n)$. We then have

 

$$\bigl\langle \tilde f(\vec\theta) \bigr\rangle \frac{1}{A^N} \int_\Omega \rm d^2 \theta_1 \int_\Omega \rm d^2 \t... ...( \vec\theta - \vec\theta_n)}{\sum_{n=1}^N w( \vec\theta - \vec\theta_n )}\cdot$$ (7)

Relabeling the integration variables we can rewrite this expression as

 

$$\bigl\langle \tilde f(\vec\theta) \bigr\rangle \frac{N}{A^N} \int_\Omega \rm d^2 \theta_1 \int_\Omega \rm d^2 \t... ...( \vec\theta - \vec\theta_1)}{\sum_{n=1}^N w( \vec\theta - \vec\theta_n )}\cdot$$ (8)

We now define a new random variable

$$y(\vec\theta) = \sum_{n=2}^N w( \vec\theta - \vec\theta_n ) .$$ (9)

Note that the sum runs from n=2 to n=N. Let us call p_y(y) the probability distribution for $y(\vec\theta)$. If we suppose that $\vec\theta$ is not close to the boundary of $\Omega$, so that the support of $w(\vec\theta - \vec\theta')$ (i.e. the set of points $\vec\theta'$ where $w(\vec\theta - \vec\theta') \neq 0$) is inside $\Omega$, then the probability distribution for $y(\vec\theta)$ does not depend on $\vec\theta$. We anticipate here that below we will take the limit of large surveys, so that $\Omega$ tends to the whole plane, and $A \rightarrow \infty$, $N \rightarrow \infty$, such that $\rho = N / A$ remains constant. Since, by definition, the weight function is assumed to be non-negative, p_y(y) vanishes for y < 0. Analogously, we call p_w(w) the probability distribution for the weight w. These two probability distributions can be calculated from the equations

  

$$\frac{1}{A} \int_\Omega \delta\bigl( w - w(\vec\theta) \bigr) \, \rm d^2 \theta ,$$ (10)

$$\frac{1}{A^{N-1}} \int_\Omega \rm d^2 \theta_2 \cdots \int_\Omega \rm d^2 \theta_N \delta( y - w_2 - \cdots - w_N ) \int_0^\infty \rm dw_2 p_w( w_2 ) \int_0^\infty \rm dw_3 p_w( w_3 ) \: \cdots \int_0^\infty \rm dw_N p_w( w_N ) \delta( y - w_2 - \cdots - w_N ) ,$$ (11)

where $\delta$ is Dirac's distribution and where we have called $w_n = w(\vec\theta_n)$. Note that Eqs. (10) and (11) hold only if the N points $\bigl\{ \vec\theta_n \bigr\}$ are uniformly distributed on the area A with density $\rho$, and if there is no correlation (so that the probability distribution for each point is $p_{\vec\theta}(\vec\theta_n) = 1/A$). Moreover, we are assuming here that the probability distribution for $y(\vec\theta)$ does not depend on $\vec\theta$. This is true only if a given configuration of points $\bigl\{ \vec\theta_n \bigr\}$ has the same probability as the translated set $\bigl\{ \vec\theta_n + \vec\phi \bigr\}$. This translation invariance, clearly, cannot hold exactly for finite fields $\Omega$; on the other hand, again, as long as $\vec\theta$ is far from the boundary of the field, the probability distribution for $y(\vec\theta)$ is basically independent of $\vec\theta$. Note that in the case of a field with masks, we also have to exclude in our analysis points close to the masks.

Using p_y we can rewrite Eq. (8) in a more compact form:

 

$$\bigl\langle \tilde f(\vec\theta) \bigr\rangle \rho \int_\Omega \rm d^2 \theta_1 f( \vec\theta_1 ) w( \vec\theta... ...ta_1) \int_0^\infty \frac{p_y(y)}{w(\vec\theta - \vec\theta_1) + y} \, \rm dy ,$$ (12)

where, we recall, $\rho = N / A$ is the density of objects. For the following calculations, it is useful to write this equation as

$$\bigl\langle \tilde f(\vec\theta) \bigr\rangle = \int_\Om... ...bigl( w(\vec\theta - \vec\theta') \bigr) \, \rm d^2 \theta' ,$$ (13)

where C(w) the correcting factor, defined as

$$C(w) = \rho \int_0^\infty \frac{p_y(y)}{w + y} \, \rm dy .$$ (14)

Finally, we will often call the combination $w_{\rm eff}(\vec\theta) = w(\vec\theta) C\bigl( w(\vec\theta) \bigr)$, which enters Eq. (13), effective weight.

Interestingly, Eq. (13) shows that the relationship between $\bigl\langle \tilde f(\vec\theta) \bigr\rangle$ and $f(\vec\theta)$is a simple convolution with the kernel $w_{\rm eff}$. From the definition (5), we can also see that this kernel is normalized, in the sense that

$$\int_\Omega \bigl\langle \tilde f(\vec\theta) \bigr\rangle ... ...rm d^2 \theta = \int_\Omega f(\vec\theta) \, \rm d^2 \theta .$$ (15)

In fact, if we consider a "flat'' signal, for instance $f(\vec\theta) = 1$, we clearly obtain $\bigl\langle \tilde f(\vec\theta) \bigr\rangle = 1$. On the other hand, from the properties of convolutions, we know that the ratio between the l.h.s. and the r.h.s. of Eq. (15) is constant, independent of the function $f(\vec\theta)$. We thus deduce that this ratio is 1, i.e. that Eq. (15) holds in general. The normalization of $w_{\rm eff}$ will be also proved below in Sect. 5.1 using analytical techniques.

If p_y(y) is available, Eq. (12) can be used to obtain the expectation value for the smoothed map $\tilde f$. In order to obtain an expression for p_y we use Markov's method (see, e.g., Chandrasekhar 1943; see also Deguchi & Watson 1987 for an application to microlensing studies). Let us define the Laplace transforms of p_y and p_w:

  

$$\mathcal L[p_w](s) = \int_0^\infty \rm e^{-sw} p_w(w) \, \rm dw =\frac{1}{A} \int_\Omega \rm e^{-sw(\vec\theta)} \, \rm d^2 \theta ,$$ (16)

$$\mathcal L[p_y](s) = \int_0^\infty \rm e^{-sy} p_y(y) \, \rm dy = \bigl[ W(s) \bigr]^{N-1} .$$ (17)

Hence p_y can in principle be obtained from the following scheme. First, we evaluate W(s) using Eq. (16), then we calculate Y(s) from Eq. (17), and finally we back-transform this function to obtain p_y(y).