Polynomial Approximations of Conditional Expectations in Scalar Gaussian Channels

We consider a channel
$Y=X+N$
where
$X$
is a random variable satisfying
$\mathbb{E}[\vert X\vert] < \infty$
and
$N$
is an independent standard normal random variable. We show that the minimum mean-square estimator of
$X$
from
$Y$
, which is given by the conditional expectation
$\mathbb{E}[X\vert Y]$
, is a polynomial in
$Y$
if and only if it is linear or constant; these two cases correspond to
$X$
being Gaussian or a constant, respectively. We also prove that the higher-order derivatives of
$y\rightarrow \mathbb{E}[X\vert Y=y]$
are expressible as multivariate polynomials in the functions
$y\rightarrow \mathbb{E}[(X-\mathbb{E}[X\vert Y])^{k}-\vert Y=y]$
for
$k\in \mathbb{N}$
. These expressions yield bounds on the 2-norm of the derivatives of the conditional expectation. These bounds imply that, if
$X$
has a compactly-supported density that is even and decreasing on the positive half-line, then the error in approximating the conditional expectation
$\mathbb{E}[X\vert Y]$
by polynomials in
$Y$
of degree at most
$n$
decays faster than any polynomial in
$n$
.