(74)

Here, I have used the property that the ?i’s are independent random variables and are

also uncorrelated with the coefficients bi. Obviously, for the smaller ?i’s, the relative

contribution of s N 2 is larger. Basically, the problem arises because, regardless of the

magnitude of ?i, the noise variance is the same in every component. If we know that there is some number h such that, for every i>h, the coefficient bi is negligible, then

the series can be truncated at i=h, and that will limit the noise. The total noise

contribution to the variance of will be

Note that, in most cases, this series does not converge as h? 8 . So how should we estimate f(y)? Let {?i} be a set of coefficients to be determined,

and let

where . The error is

Using once again the orthonormality of the eigenfunctions,

Now

and

since bi is independent of ?j. So