Wavelet-based estimation of the derivatives of a function from a heteroscedastic multichannel convolution model

We observe $n$ heteroscedastic stochastic processes where, for any $v\in\{1,...,n\}$, a convolution product of an unknown function $f$ and a known function $g_v$ is corrupted by Gaussian noise. Under a particular ordinary smooth assumption on $g_1,...,g_n$, we aim to estimate the $d$-th derivatives of $f$ from the observations. We consider an adaptive estimator based on a particular wavelet block thresholding: the "BlockJS estimator". Taking the mean integrated squared error (MISE), we prove that it achieves near optimal rates of convergence over a wide range of smoothness classes. The theory is illustrated with some numerical examples. Performance comparisons with some others methods existing in the literature are provided.