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Fast and backward stable transforms between spherical harmonic expansions and bivariate Fourier series. (arXiv:1705.05448v1 [math.NA])
来源于:arXiv
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 $\mathcal{O}(n^3\log n)$
flops; and, asymptotically optimal execution time of $\mathcal{O}(n^2\log^2 n)$
is rigorously proved via connection to Fourier integral operators. 查看全文>>