Abstract
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 .