关于球谐展开式的收敛域
On the domain of convergence of spherical harmonic expansions
摘要：
球谐展开在大多数物理科学中，特别是在物理大地测量中起着重要的作用。尽管几十年的研究，大的顺序行为的SHE系数，和精确的域收敛这些扩展，仍然是悬而未决的问题。这些问题在本文中解决的一般行星，其形状（地形）可能包括许多当地的高峰，但只有一个全球最高峰。我们表明，无论密度和地形的光滑度，缺乏彻底的解析性，球谐展开的引力势收敛正好在封闭的布里渊球（最小的球体周围的质量中心的行星包含在其内部的行星）的外部，和布里渊球以下的收敛发生概率为零。更准确地说，这种过收敛发生在参数空间中的零测度集上。一个相关的结果是，在一个自然的巴拿赫空间，SHE收敛的潜在的布里渊球以下的势函数发生在一个子空间的无限余维（而任何积极的余维已经意味着发生的概率为零）。在傅立叶空间中存在一定极限的条件下，我们发现了SHE系数的首阶渐近行为。我们更进一步，找到了布里渊球以下收敛的充要条件，这需要最高峰处的解析性形式，但这对于现实天体来说是不成立的。也就是说，布里渊球上的调和测度的纵向平均值在与行星边界的切触点处必须是实解析的。事实证明，只有一个小的附近的峰值是在这种情况下。
英文摘要：
Spherical harmonic expansions (SHEs) play an important role in most of the physical sciences, especially in physical geodesy. Despite many decades of investigation, the large order behavior of the SHE coefficients, and the precise domain of convergence for these expansions, have remained open questions. These questions are settled in the present paper for generic planets, whose shape (topography) may include many local peaks, but just one globally highest peak. We show that regardless of the smoothness of the density and topography, short of outright analyticity, the spherical harmonic expansion of the gravitational potential converges exactly in the closure of the exterior of the Brillouin sphere (The smallest sphere around the center of mass of the planet containing the planet in its interior), and convergence below the Brillouin sphere occurs with probability zero. More precisely, such over-convergence occurs on zero measure sets in the space of parameters. A related result is that, in a natural Banach space, SHE convergence of the potential below the Brillouin sphere occurs for potential functions in a subspace of infinite codimension (while any positive codimension already implies occurrence of probability zero). Provided a certain limit in Fourier space exists, we find the leading order asymptotic behavior of the coefficients of SHEs. We go further by finding a necessary and sufficient condition for convergence below the Brillouin sphere, which requires a form of analyticity at the highest peak, which would not hold for a realistic celestial body. Namely, a longitudinal average of the harmonic measure on the Brillouin sphere would have to be real-analytic at the point of contact with the boundary of the planet. It turns out that only a small neighborhood of the peak is involved in this condition.
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