  We consider the problem of estimating the unknown response function in the
multichannel deconvolution model with long-range dependent Gaussian errors. We
do not limit our consideration to a specific type of long-range dependence
rather we assume that the errors should satisfy a general assumption in terms
of the smallest and larger eigenvalues of their covariance matrices. We derive
minimax lower bounds for the quadratic risk in the proposed multichannel
deconvolution model when the response function is assumed to belong to a Besov
ball and the blurring function is assumed to possess some smoothness
properties, including both regular-smooth and super-smooth convolutions.
Furthermore, we propose an adaptive wavelet estimator of the response function
that is asymptotically optimal (in the minimax sense), or near-optimal within a
logarithmic factor, in a wide range of Besov balls. It is shown that the
optimal convergence rates depend on the balance between the smoothness
parameter of the response function, the kernel parameters of the blurring
function, the long memory parameters of the errors, and how the total number of
observations is distributed among the total number of channels. Some examples
of inverse problems in mathematical physics where one needs to recover initial
or boundary conditions on the basis of observations from a noisy solution of a
partial differential equation are used to illustrate the application of the
theory we developed. The optimal convergence rates and the adaptive estimators
we consider extend the ones studied by Pensky and Sapatinas (2009, 2010) for
independent and identically distributed Gaussian errors to the case of
long-range dependent Gaussian errors.
