  We consider the problem of estimating an input signal from noisy measurements
in both parallel scalar Gaussian channels and linear mixing systems. The
performance of the estimation process is quantified by the $\ell_\infty$ norm
error metric. We first study the minimum mean $\ell_\infty$ error estimator in
parallel scalar Gaussian channels, and verify that, when the input is
independent and identically distributed (i.i.d.) mixture Gaussian, the Wiener
filter is asymptotically optimal with probability 1. For linear mixing systems
with i.i.d. sparse Gaussian or mixture Gaussian inputs, under the assumption
that the relaxed belief propagation (BP) algorithm matches Tanaka's fixed point
equation, applying the Wiener filter to the output of relaxed BP is also
asymptotically optimal with probability 1. However, in order to solve the
practical problem where the signal dimension is finite, we apply an estimation
algorithm that has been proposed in our previous work, and illustrate that an
$\ell_\infty$ error minimizer can be approximated by an $\ell_p$ error
minimizer provided the value of $p$ is properly chosen.
