2-FAST：Fast和投影两点函数的精确计算
2-FAST: Fast and accurate computation of projected two-point functions
摘要：
本文提出了一种快速精确的球面贝塞尔变换（2-FAST）算法，该算法可以快速精确地计算包含一个或两个球面贝塞尔函数的积分。这些类型的积分发生在将星系功率谱$P(k)$投影到位形空间$ξ_\ell^ν(r)$或球谐空间$C_\ell(χ,χ')$上时。首先，我们采用的FFT对数变换的功率谱的计算分为$P(k)$相关的系数和$P(k)$无关的基础函数乘以球面贝塞尔函数的积分。我们发现解析表达式为后者积分的特殊功能，递归提供了一个快速和准确的评价。因此，该算法避免了直接积分高度振荡的球面贝塞尔函数。
英文摘要：
We present the 2-point function from Fast and Accurate Spherical Bessel Transformation (2-FAST) algorithm for a fast and accurate computation of integrals involving one or two spherical Bessel functions. These types of integrals occur when projecting the galaxy power spectrum $P(k)$ onto the configuration space, $ξ_\ell^ν(r)$, or spherical harmonic space, $C_\ell(χ,χ')$. First, we employ the FFTlog transformation of the power spectrum to divide the calculation into $P(k)$-dependent coefficients and $P(k)$-independent integrations of basis functions multiplied by spherical Bessel functions. We find analytical expressions for the latter integrals in terms of special functions, for which recursion provides a fast and accurate evaluation. The algorithm, therefore, circumvents direct integration of highly oscillating spherical Bessel functions.
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