Fast and backward stable transforms between spherical harmonic expansions and bivariate Fourier series
Fast and backward stable transforms between spherical harmonic expansions and bivariate Fourier series
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DOI：
10.1016/j.acha.2017.11.001
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发表时间：
2017-05
影响因子：
2.5
通讯作者：
R. Slevinsky
中科院分区：
文献类型：
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作者：
R. Slevinsky
在二元傅立叶级数中球谐展开式和它们的类似物之间导出快速变换。基的变化分两步描述：首先，将所有阶归一化关联勒让德函数的展开转换为零阶和一阶的展开；然后，这些中间表达式以三角形式重新展开。第一步对连接系数的良条件矩阵进行蝶式分解。第二步通过分层非对角低秩矩阵分解进行快速正交多项式变换。总的预计算最多需要 O(n 3 log n) 次触发器；并且，通过与傅立叶积分算子的连接，严格证明了 O (n 2 log 2 n) 的渐近最优执行时间。
A rapid transformation is derived between spherical harmonic expansions and their analogues in a bivariate Fourier series. The change of basis is described in two steps: firstly, expansions in normalized associated Legendre functions of all orders are converted to those of order zero and one; then, these intermediate expressions are re-expanded in trigonometric form. The first step proceeds with a butterfly factorization of the well-conditioned matrices of connection coefficients. The second step proceeds with fast orthogonal polynomial transforms via hierarchically off-diagonal low-rank matrix decompositions. Total pre-computation requires at best O (n 3 log n) flops; and, asymptotically optimal execution time of O (n 2 log 2 n) is rigorously proved via connection to Fourier integral operators.