  Uniform asymptotic confidence bands for a multivariate regression function in
an inverse regression model with a convolution-type operator are constructed.
The results are derived using strong approximation methods and a limit theorem
for the supremum of a stationary Gaussian field over an increasing system of
sets. As a particular application, asymptotic confidence bands for a time
dependent regression function $f_t(x)$ ($x\in \mathbb {R}^d,t\in \mathbb {R}$)
in a convolution-type inverse regression model are obtained. Finally, we
demonstrate the practical feasibility of our proposed methods in a simulation
study and an application to the estimation of the luminosity profile of the
elliptical galaxy NGC5017. To the best knowledge of the authors, the results
presented in this paper are the first which provide uniform confidence bands
for multivariate nonparametric function estimation in inverse problems.
