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Polynomial Approximations of Conditional Expectations in Scalar Gaussian Channels

We consider a channel $Y=X+N$ where $X$ is a random variable satisfying $\mathbb{E}[|X|]<\infty$ and $N$ is an independent standard normal random variable. We show that the minimum mean-square error estimator of $X$ from $Y,$ which is given by the conditional expectation $\mathbb{E}[X \mid Y],$ 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 $y \mapsto \mathbb{E}[X \mid Y=y]$ are expressible as multivariate polynomials in the functions $y \mapsto \mathbb{E}\left[ \left( X - \mathbb{E}[X \mid Y] \right)^k \mid Y = y \right]$ 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 $\mathbb{E}[X \mid Y]$ by polynomials in $Y$ of degree at most $n$ decays faster than any polynomial in $n.$