Abstract The main focus of this paper is on studying an order optimal regularization scheme based on the Meyer wavelets method to solve the analytic continuation problem in the high-dimensional complex domain Ω : = { x + i y ∈ C N : x ∈ R N , ‖ y ‖ ≤ ‖ y 0 ‖ , y , y 0 ∈ R + N } . This problem is exponentially ill-posed and suffers from the Hadamard's instability. Theoretically, we first provide an optimal conditional stability estimate for the proposed original problem. Applying the Meyer wavelets, an order optimal regularization scheme is then developed to stabilize the considered ill-posed problem. Some sharp error estimates of the Holder-Logarithmic type controlled by the Sobolev scale under an a-priori information are also derived. The provided error estimates are of the order optimal in the sense of Tautenhahn. Finally, some different one- and two-dimensional examples are presented to confirm the efficiency and applicability of the proposed regularization scheme. The comparison results also show that the proposed method is more accurate than the other existing methods in the literature.

中文翻译：

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关于不适定解析延拓问题：一种阶次最优正则化方案

摘要 本文的主要重点是研究一种基于Meyer小波方法的阶次最优正则化方案来解决高维复域Ω中的解析延拓问题：= { x + iy ∈ CN : x ∈ RN , ‖ y ‖ ≤ ‖ y 0 ‖ , y , y 0 ∈ R + N } 。这个问题是指数不适定的，并且受到 Hadamard 的不稳定性的影响。从理论上讲，我们首先为提出的原始问题提供最佳条件稳定性估计。应用迈耶小波，然后开发阶次最优正则化方案来稳定所考虑的不适定问题。还导出了在先验信息下由 Sobolev 标度控制的 Holder-Logarithmic 类型的一些尖锐误差估计。提供的误差估计是 Tautenhahn 意义上的最优阶。最后，提供了一些不同的一维和二维示例，以确认所提出的正则化方案的效率和适用性。比较结果还表明，所提出的方法比文献中的其他现有方法更准确。