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.

中文翻译：

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具有长记忆噪声的各向异性函数去卷积：多参数分数维纳表的情况

我们研究具有两参数分数高斯噪声的各向异性二维函数去卷积模型的极小极大结果。我们推导出 -risk 的下限，并利用 Riesz 多项势，我们应用小波-暗角扩展来去相关各向异性分数高斯噪声。我们构建了一个自适应小波硬阈值估计器，它在广泛的 Besov 球中获得渐近最优或准最优收敛率。这种收敛速度取决于 Besov 球的参数、卷积算子的不适定度和规则平滑卷积下的分数高斯噪声参数之间的微妙平衡，而该速度不受长记忆的影响在超平滑卷积下。有限的模拟研究证实了论文的理论主张。所提出的方法扩展到一般的 r 维情况，r>2，并且相应的收敛速度不受维数灾难的影响。