  We consider the statistical deconvolution problem where one observes $n$
replications from the model $Y=X+\epsilon$, where $X$ is the unobserved random
signal of interest and $\epsilon$ is an independent random error with
distribution $\phi$. Under weak assumptions on the decay of the Fourier
transform of $\phi,$ we derive upper bounds for the finite-sample sup-norm risk
of wavelet deconvolution density estimators $f_n$ for the density $f$ of $X$,
where $f:\mathbb{R}\to \mathbb{R}$ is assumed to be bounded. We then derive
lower bounds for the minimax sup-norm risk over Besov balls in this estimation
problem and show that wavelet deconvolution density estimators attain these
bounds. We further show that linear estimators adapt to the unknown smoothness
of $f$ if the Fourier transform of $\phi$ decays exponentially and that a
corresponding result holds true for the hard thresholding wavelet estimator if
$\phi$ decays polynomially. We also analyze the case where $f$ is a
"supersmooth"/analytic density. We finally show how our results and recent
techniques from Rademacher processes can be applied to construct global
confidence bands for the density $f$.
