Anisotropic functional deconvolution with long-memory noise: the case of a multi-parameter fractional Wiener sheet

We look into the minimax results for the anisotropic two-dimensional functional deconvolution model with the two-parameter fractional Gaussian noise. We derive the lower bounds for the -risk, , and taking advantage of the Riesz poly-potential, we apply a wavelet-vaguelette expansion to de-correlate the anisotropic fractional Gaussian noise. We construct an adaptive wavelet hard-thresholding estimator that attains asymptotically optimal or quasi-optimal convergence rates in a wide range of Besov balls. Such convergence rates depend on a delicate balance between the parameters of the Besov balls, the degree of ill-posedness of the convolution operator and the parameters of the fractional Gaussian noise under regular-smooth convolution, whereas the rates are not affected by long-memory under super-smooth convolution. A limited simulations study confirms the theoretical claims of the paper. The proposed approach is extended to the general r-dimensional case, with r>2, and the corresponding convergence rates do not suffer from the curse of dimensionality.