Abstract

Anisotropic functional deconvolution model is investigated in the bivariate case when the design points t_i, $i=1,2,\cdots ,N,$ and x_l, $l=1,2,\cdots ,M,$ are irregular and follow known densities h₁, h₂, respectively. In particular, we focus on the case when the densities h₁ and h₂ have singularities, but $1/h_1$ and $1/h_2$ are still integrable on [0, 1]. We construct an adaptive wavelet estimator that attains asymptotically near-optimal convergence rates in a wide range of Besov balls. The convergence rates are completely new and depend on a balance between the smoothness and the spatial homogeneity of the unknown function f, the degree of ill-posed-ness of the convolution operator and the degrees of spatial irregularity associated with h₁ and h₂. Nevertheless, the spatial irregularity affects convergence rates only when f is spatially inhomogeneous in either direction.

摘要

各向异性函数反卷积模型在设计点t _i时的双变量情况下进行研究，$\mathrm{一世}=1,2,\cdots ,ñ,$和xl , _{_}$l=1,2,\cdots ,米,$是不规则的并且分别遵循已知的密度h ₁、h ₂。特别是，我们关注密度h ₁和h ₂具有奇点的情况，但是$1/H_1$和$1/H_2$在 [0, 1] 上仍然是可积的。我们构建了一个自适应小波估计器，该估计器在各种 Besov 球中获得渐近接近最优的收敛速度。收敛速度是全新的，取决于未知函数f的平滑度和空间同质性、卷积算子的不适定度以及与h ₁和h ₂相关的空间不规则度之间的平衡。然而，只有当f在任一方向上空间不均匀时，空间不规则性才会影响收敛速度。