亚纯延拓中可见极点的行列式特征与辐角原理认证
Determinant Characteristics and Argument-Principle Certification for Visible Poles in Meromorphic Continuation
AI总结：
研究单位圆盘内从圆形边界数据向外的亚纯延拓，引入移位行列式特征，证明纯有限极点模型中行列式性质，提出根提议和轮廓认证程序，给出稳定性条件并通过实验验证相关结论。
摘要：
我们研究单位圆盘内从圆形边界数据向外的亚纯延拓。未知函数在单位圆盘内全纯，并允许亚纯延拓到一个更大的圆盘，其中有限多个外部单极点叠加在未知的全纯背景上。边界迹的正傅里叶系数是原点处的泰勒系数，外部极点在这些系数中产生有限的指数和分量。我们引入移位行列式特征，并证明在纯有限极点模型中，正确阶数的行列式恰好分解为一个非零常数乘以其零点为外部极点倒数的多项式。对于无噪声等距离散傅里叶系数，得到相同的零集；采样仅通过混叠因子改变幅度。对于包含全纯背景、离散化效应和噪声的数据，单个经验行列式的根仅是候选倒数极点。因此，我们提出一种根提议和轮廓认证程序：行列式根生成候选区域，而局部辐角原理计数、轮廓矩、经验余量以及行列式阶数和移位的持续性认证可见极点。一种鲁歇型扰动分析给出稳定局部零计数的充分条件，并解释留数、极点间距、到目标圆环边界的距离、移位和噪声如何影响可见性。数值实验验证了纯极点恒等式，展示了在背景和噪声下的认证，并表明高噪声、弱留数、靠近边界的极点和靠近的极点自然会导致轮廓认证可见极点的部分恢复。
英文摘要：
We study outward meromorphic continuation from circular boundary data in the unit disk. The unknown function is holomorphic in the unit disk and admits a meromorphic continuation to a larger disk, where finitely many exterior simple poles are superposed on an unknown holomorphic background. The positive Fourier coefficients of the boundary trace are Taylor coefficients at the origin, and exterior poles generate a finite exponential-sum component in these coefficients. We introduce shifted determinant characteristics and prove that, in the pure finite-pole model, the determinant for the correct order factors exactly into a nonzero constant times the polynomial whose zeros are the reciprocals of the exterior poles. The same zero set is obtained for noiseless equispaced discrete Fourier coefficients; sampling changes only the amplitudes through an aliasing factor. For data containing a holomorphic background, discretization effects, and noise, roots of a single empirical determinant are only candidate reciprocal poles. We therefore propose a root-propose and contour-certify procedure: determinant roots generate candidate regions, while local argument-principle counts, contour moments, empirical margins, and persistence over determinant orders and shifts certify visible poles. A Rouché-type perturbation analysis gives sufficient conditions for stable local zero counts and explains how residues, pole separation, distance to the target annulus boundary, shifts, and noise affect visibility. Numerical experiments verify the pure-pole identity, demonstrate certification under background and noise, and show that high noise, weak residues, boundary-near poles, and close poles naturally lead to partial recovery of contour-certified visible poles.
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