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FastTransforms
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FastTransforms
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What FastTransforms actually computes.
\( \def\ud{{\rm\,d}} \def\ii{{\rm i}} \def\fl{{\rm\,fl}} \def\abs#1{\left|{#1}\right|} \def\norm#1{\left\|{#1}\right\|} \def\conj#1{\overline{#1}} \)
FastTransforms provides computational kernels and driver routines for multivariate orthogonal polynomial transforms. The algorithms are \(2\)-normwise backward stable with a runtime complexity of \(\mathcal{O}(n^{d+1})\), where \(n\) is the polynomial degree and \(d\) is the spatial dimension of the problem.
The transforms are separated into computational kernels that offer SSE, AVX, and AVX-512 vectorization on applicable Intel processors, and driver routines that are easily parallelized by OpenMP.
For every subsection below, the title of the subsection, of the form a2b, refers conceptually to the transform and the available functions are as follows:
ft_plan_a2b, is a pre-computation,ft_execute_a2b, is a forward execution,ft_execute_b2a, is a backward execution,ft_execute_a_hi2lo, is a conversion to a tensor-product basis,ft_execute_a_lo2hi, is a conversion from a tensor-product basis,ft_kernel_a_hi2lo, is an orthonormal conversion from high to low order,ft_kernel_a_lo2hi, is an orthonormal conversion from low to high order.The ft_execute_* functions are drivers that perform transforms as defined below. They are composed of computational kernels, of the form ft_kernel_*, that are designed to be sufficiently generic to be assembled for different parallelism and compute paradigms. For good use of SIMD, the computational kernels rely on row-major ordering of the coefficients, and for trivial shared memory parallelism, the driver routines rely on column-major ordering of the coefficients. Local permutations are utilized to balance the use of SIMD and OpenMP multithreading. Generally, the pre-computations construct a ft_harmonic_plan, which may be destroyed by a call to ft_destroy_harmonic_plan.
Real orthonormal spherical harmonics in \(L^2(\mathbb{S}^2)\) are:
\begin{align*} Y_\ell^m(\theta,\varphi) & = (-1)^{\abs{m}}\sqrt{(\ell+\tfrac{1}{2})\frac{(\ell-\abs{m})!}{(\ell+\abs{m})!}} P_\ell^{\abs{m}}(\cos\theta) \sqrt{\frac{2-\delta_{m,0}}{2\pi}} \left\{\begin{array}{ccc} \cos(m\varphi) & {\rm for} & m \ge 0,\\ \sin(\abs{m}\varphi) & {\rm for} & m < 0,\end{array}\right. \end{align*}
where \(P_\ell^m(\cos\theta)\) are the associated Legendre functions. For \(0\le m\le\ell\), these may be defined by differentiating Legendre polynomials:
\[ P_\ell^m(x) = (-1)^m (1-x^2)^{\frac{m}{2}} \dfrac{{\rm d}^m}{{\rm d} x^m} P_\ell(x), \]
and defined equivalently by ultraspherical polynomials:
\[ P_\ell^m(\cos\theta) = (-2)^m(\tfrac{1}{2})_m \sin^m\theta C_{\ell-m}^{(m+\frac{1}{2})}(\cos\theta), \]
so that the spherical harmonics have positive real phase. A degree- \((n-1)\) expansion in spherical harmonics is given by:
\[ f_{n-1}(\theta,\varphi) = \sum_{\ell=0}^{n-1}\sum_{m=-\ell}^{+\ell} f_\ell^m Y_\ell^m(\theta,\varphi). \]
If real spherical harmonic expansion coefficients are organized into the array:
\[ F = \begin{pmatrix} f_0^0 & f_1^{-1} & f_1^1 & f_2^{-2} & f_2^2 & \cdots & f_{n-1}^{1-n} & f_{n-1}^{n-1}\\ f_1^0 & f_2^{-1} & f_2^1 & f_3^{-2} & f_3^2 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ f_{n-3}^0 & f_{n-2}^{-1} & f_{n-2}^1 & f_{n-1}^{-2} & f_{n-1}^2 & & \vdots & \vdots\\ f_{n-2}^0 & f_{n-1}^{-1} & f_{n-1}^1 & 0 & 0 & \cdots & 0 & 0\\ f_{n-1}^0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ \end{pmatrix}, \]
then ft_plan_sph2fourier creates the appropriate ft_harmonic_plan, and ft_execute_sph2fourier returns the bivariate Fourier coefficients:
\[ G = \begin{pmatrix} g_0^0 & g_0^{-1} & g_0^1 & \cdots & g_0^{1-n} & g_0^{n-1}\\ g_1^0 & g_1^{-1} & g_1^1 & \cdots & g_1^{1-n} & g_1^{n-1}\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ g_{n-2}^0 & g_{n-2}^{-1} & g_{n-2}^1& \cdots & g_{n-2}^{1-n} & g_{n-2}^{n-1}\\ g_{n-1}^0 & 0 & 0 & \cdots & g_{n-1}^{1-n} & g_{n-1}^{n-1}\\ \end{pmatrix}. \]
That is:
\[ f_n(\theta,\varphi) = \sum_{\ell=0}^{n-1}\sum_{m=1-n}^{n-1} g_\ell^m \left\{\begin{array}{ccc} \cos(\ell\theta) & {\rm for} & m~{\rm even},\\ \sin((\ell+1)\theta) & {\rm for} & m~{\rm odd},\end{array}\right\} \sqrt{\frac{2-\delta_{m,0}}{2\pi}}\left\{\begin{array}{ccc} \cos(m\varphi) & {\rm for} & m \ge 0,\\ \sin(\abs{m}\varphi) & {\rm for} & m < 0,\end{array}\right. \]
and ft_execute_fourier2sph converts them back.
Real orthonormal generalized Zernike polynomials in \(L^2(\mathbb{D}^2, r^{2\alpha+1}(1-r^2)^\beta\ud r\ud\theta)\) are:
\[ Z_{\ell,m}^{(\alpha,\beta)}(r,\theta) = 2^{\frac{\abs{m}+\alpha+\beta+2}{2}} r^{\abs{m}}\tilde{P}_{\frac{\ell-\abs{m}}{2}}^{(\beta,\abs{m}+\alpha)}(2r^2-1) \sqrt{\frac{2-\delta_{m,0}}{2\pi}} \left\{\begin{array}{ccc} \cos(m\theta) & {\rm for} & m \ge 0,\\ \sin(\abs{m}\theta) & {\rm for} & m < 0,\end{array}\right. \]
where the tilde implies that the univariate Jacobi polynomials are orthonormal. A degree- \((2n-2)\) expansion in generalized Zernike polynomials is given by:
\[ f_{2n-2}(r,\theta) = \sum_{\ell=0}^{2n-2}\sum_{m=-\ell,2}^{+\ell} f_\ell^m Z_{\ell,m}^{(\alpha,\beta)}(r,\theta), \]
where the \(,2\) in the inner summation index implies that the inner summation runs from \(m=-\ell\) in steps of \(2\) up to \(+\ell\). If generalized Zernike expansion coefficients are organized into the array:
\[ F = \begin{pmatrix} f_0^0 & f_1^{-1} & f_1^1 & f_2^{-2} & f_2^2 & \cdots & f_{2n-2}^{2-2n} & f_{2n-2}^{2n-2}\\ f_2^0 & f_3^{-1} & f_3^1 & f_4^{-2} & f_4^2 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ f_{2n-6}^0 & f_{2n-5}^{-1} & f_{2n-5}^1 & f_{2n-4}^{-2} & f_{2n-4}^2 & & \vdots & \vdots\\ f_{2n-4}^0 & f_{2n-3}^{-1} & f_{2n-3}^1 & f_{2n-2}^{-2} & f_{2n-2}^2 & \cdots & 0 & 0\\ f_{2n-2}^0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ \end{pmatrix}, \]
then ft_plan_disk2cxf creates the appropriate ft_harmonic_plan, and ft_execute_disk2cxf returns the even Chebyshev–Fourier coefficients:
\[ G = \begin{pmatrix} g_0^0 & g_0^{-1} & g_0^1 & g_0^{-2} & g_0^2 & \cdots & g_0^{2-2n} & g_0^{2n-2}\\ g_2^0 & g_2^{-1} & g_2^1 & g_2^{-2} & g_2^2 & \cdots & g_2^{2-2n} & g_2^{2n-2}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ g_{2n-4}^0 & g_{2n-4}^{-1} & g_{2n-4}^1 & g_{2n-4}^{-2} & g_{2n-4}^2 & \cdots & g_{2n-4}^{2-2n} & g_{2n-4}^{2n-2}\\ g_{2n-2}^0 & 0 & 0 & g_{2n-2}^{-2} & g_{2n-2}^2 & \cdots & g_{2n-2}^{2-2n} & g_{2n-2}^{2n-2}\\ \end{pmatrix}. \]
That is:
\[ f_{2n-2}(r,\theta) = \sum_{\ell=0}^{n-1}\sum_{m=2-2n}^{2n-2} g_{2\ell}^m \left\{\begin{array}{ccc} T_{2\ell}(r) & {\rm for} & m~{\rm even},\\ T_{2\ell+1}(r) & {\rm for} & m~{\rm odd},\end{array}\right\} \sqrt{\frac{2-\delta_{m,0}}{2\pi}}\left\{\begin{array}{ccc} \cos(m\theta) & {\rm for} & m \ge 0,\\ \sin(\abs{m}\theta) & {\rm for} & m < 0,\end{array}\right. \]
and ft_execute_cxf2disk converts them back.
Consider \(P_n^{t,(\alpha,\beta,\gamma)}(x)\) to be orthogonal polynomials in \(L^2([-1,1], (1-x)^\alpha(1+x)^\beta(t+x)^\gamma\ud x)\), for parameter values \(\{t>1, \alpha,\beta>-1, \gamma\in\mathbb{R}\}\cup\{t=1, \alpha,\beta+\gamma>-1\}\).
Real orthonormal annulus polynomials in \(L^2(\{(r,\theta) : \rho < r < 1, 0 < \theta < 2\pi\}, r^{2\gamma+1}(r^2-\rho^2)^\alpha(1-r^2)^\beta\ud r\ud\theta)\) are:
\[ Z_{\ell,m}^{\rho,(\alpha,\beta,\gamma)}(r,\theta) = \sqrt{2} \left(\frac{2}{1-\rho^2}\right)^{\frac{\abs{m}+\alpha+\beta+\gamma+1}{2}} r^{\abs{m}}\tilde{P}_{\frac{\ell-\abs{m}}{2}}^{\frac{1+\rho^2}{1-\rho^2},(\beta,\alpha,\abs{m}+\gamma)}\left(\frac{2r^2-1-\rho^2}{1-\rho^2}\right) \sqrt{\frac{2-\delta_{m,0}}{2\pi}} \left\{\begin{array}{ccc} \cos(m\theta) & {\rm for} & m \ge 0,\\ \sin(\abs{m}\theta) & {\rm for} & m < 0,\end{array}\right. \]
where the tilde implies that \(\tilde{P}_n^{t,(\alpha,\beta,\gamma)}(x)\) are orthonormal. In the limit as \(\rho\to0\), the annulus polynomials converge to the Zernike polynomials, \(Z_{\ell,m}^{0,(\alpha,\beta,\gamma)}(r, \theta) = Z_{\ell,m}^{(\alpha+\gamma,\beta)}(r, \theta)\). A degree- \((2n-2)\) expansion in annulus polynomials is given by:
\[ f_{2n-2}(r,\theta) = \sum_{\ell=0}^{2n-2}\sum_{m=-\ell,2}^{+\ell} f_\ell^m Z_{\ell,m}^{\rho,(\alpha,\beta,\gamma)}(r,\theta), \]
where the \(,2\) in the inner summation index implies that the inner summation runs from \(m=-\ell\) in steps of \(2\) up to \(+\ell\). If annulus polynomial expansion coefficients are organized into the array:
\[ F = \begin{pmatrix} f_0^0 & f_1^{-1} & f_1^1 & f_2^{-2} & f_2^2 & \cdots & f_{2n-2}^{2-2n} & f_{2n-2}^{2n-2}\\ f_2^0 & f_3^{-1} & f_3^1 & f_4^{-2} & f_4^2 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ f_{2n-6}^0 & f_{2n-5}^{-1} & f_{2n-5}^1 & f_{2n-4}^{-2} & f_{2n-4}^2 & & \vdots & \vdots\\ f_{2n-4}^0 & f_{2n-3}^{-1} & f_{2n-3}^1 & f_{2n-2}^{-2} & f_{2n-2}^2 & \cdots & 0 & 0\\ f_{2n-2}^0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ \end{pmatrix}, \]
then ft_plan_ann2cxf creates the appropriate ft_harmonic_plan, and ft_execute_ann2cxf returns the even Chebyshev–Fourier coefficients:
\[ G = \begin{pmatrix} g_0^0 & g_0^{-1} & g_0^1 & g_0^{-2} & g_0^2 & \cdots & g_0^{2-2n} & g_0^{2n-2}\\ g_2^0 & g_2^{-1} & g_2^1 & g_2^{-2} & g_2^2 & \cdots & g_2^{2-2n} & g_2^{2n-2}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ g_{2n-4}^0 & g_{2n-4}^{-1} & g_{2n-4}^1 & g_{2n-4}^{-2} & g_{2n-4}^2 & \cdots & g_{2n-4}^{2-2n} & g_{2n-4}^{2n-2}\\ g_{2n-2}^0 & 0 & 0 & g_{2n-2}^{-2} & g_{2n-2}^2 & \cdots & g_{2n-2}^{2-2n} & g_{2n-2}^{2n-2}\\ \end{pmatrix}. \]
That is:
\[ f_{2n-2}(r,\theta) = \sum_{\ell=0}^{n-1}\sum_{m=2-2n}^{2n-2} g_{2\ell}^m \left\{\begin{array}{ccc} T_{\ell}\left(\frac{2r^2-1-\rho^2}{1-\rho^2}\right) & {\rm for} & m~{\rm even},\\ \sqrt{\frac{2}{1-\rho^2}}rT_{\ell}\left(\frac{2r^2-1-\rho^2}{1-\rho^2}\right) & {\rm for} & m~{\rm odd},\end{array}\right\} \sqrt{\frac{2-\delta_{m,0}}{2\pi}}\left\{\begin{array}{ccc} \cos(m\theta) & {\rm for} & m \ge 0,\\ \sin(\abs{m}\theta) & {\rm for} & m < 0,\end{array}\right. \]
and ft_execute_cxf2ann converts them back.
Real orthonormal Dunkl–Xu polynomials in \(L^2(\mathbb{D}^2, (1-x^2-y^2)^\beta\ud x\ud y)\) are:
\[ \tilde{P}_{\ell,m}^{(\beta)}(x,y) = (1-x^2)^{\frac{m}{2}}\tilde{P}_{\ell-m}^{(m+\beta+\frac{1}{2},m+\beta+\frac{1}{2})}(x) \tilde{P}_m^{(\beta,\beta)}\left(\frac{y}{\sqrt{1-x^2}}\right), \]
where the tilde implies that the univariate Jacobi polynomials are orthonormal. A degree- \((n-1)\) expansion in Dunkl–Xu polynomials is given by:
\[ f_{n-1}(x,y) = \sum_{\ell=0}^{n-1}\sum_{m=0}^\ell f_\ell^m \tilde{P}_{\ell,m}^{(\beta)}(x,y). \]
If Dunkl–Xu expansion coefficients are organized into the array:
\[ F = \begin{pmatrix} f_0^0 & f_1^1 & f_2^2 & \cdots & f_{n-1}^{n-1}\\ \vdots & \vdots & \vdots & \ddots & 0\\ f_{n-3}^0 & f_{n-2}^1 & f_{n-1}^2 & & \vdots\\ f_{n-2}^0 & f_{n-1}^1 & 0 & \cdots & 0\\ f_{n-1}^0 & 0 & 0 & \cdots & 0\\ \end{pmatrix}, \]
then ft_plan_rectdisk2cheb creates the appropriate ft_harmonic_plan, and ft_execute_rectdisk2cheb returns the bivariate (partially weighted) Chebyshev coefficients:
\[ G = \begin{pmatrix} g_0^0 & g_0^1 & \cdots & g_0^{n-1}\\ g_1^0 & g_1^1 & \cdots & g_1^{n-1}\\ \vdots & \vdots & \ddots & \vdots\\ g_{n-2}^0 & g_{n-2}^1 & \cdots & g_{n-2}^{n-1}\\ g_{n-1}^0 & 0 & \cdots & g_{n-1}^{n-1}\\ \end{pmatrix}. \]
That is:
\[ f_{n-1}(x,y) = \sum_{\ell=0}^{n-1}\sum_{m=0}^{n-1} g_\ell^m \left\{\begin{array}{ccc} T_\ell(x) & {\rm for} & m~{\rm even},\\ \sqrt{1-x^2}U_\ell(x) & {\rm for} & m~{\rm odd},\end{array}\right\} T_m\left(\frac{y}{\sqrt{1-x^2}}\right), \]
and ft_execute_cheb2rectdisk converts them back.
Proriol polynomials on the triangle with vertices \((0,0)\), \((1,0)\), and \((0,1)\) are:
\[ \tilde{P}_{\ell,m}^{(\alpha,\beta,\gamma)}(x,y) = (2-2x)^m \tilde{P}_{\ell-m}^{(2m+\beta+\gamma+1,\alpha)}(2x-1) \tilde{P}_m^{(\gamma,\beta)}\left(\frac{2y}{1-x}-1\right), \]
where the tilde implies that the univariate Jacobi polynomials are orthonormal. A degree- \((n-1)\) expansion in Proriol polynomials is given by:
\[ f_{n-1}(x,y) = \sum_{\ell=0}^{n-1}\sum_{m=0}^\ell f_\ell^m \tilde{P}_{\ell,m}^{(\alpha,\beta,\gamma)}(x,y). \]
If Proriol expansion coefficients are organized into the array:
\[ F = \begin{pmatrix} f_0^0 & f_1^1 & f_2^2 & \cdots & f_{n-1}^{n-1}\\ \vdots & \vdots & \vdots & \ddots & 0\\ f_{n-3}^0 & f_{n-2}^1 & f_{n-1}^2 & & \vdots\\ f_{n-2}^0 & f_{n-1}^1 & 0 & \cdots & 0\\ f_{n-1}^0 & 0 & 0 & \cdots & 0\\ \end{pmatrix}, \]
then ft_plan_tri2cheb creates the appropriate ft_harmonic_plan, and ft_execute_tri2cheb returns the bivariate Chebyshev coefficients:
\[ G = \begin{pmatrix} g_0^0 & g_0^1 & \cdots & g_0^{n-1}\\ g_1^0 & g_1^1 & \cdots & g_1^{n-1}\\ \vdots & \vdots & \ddots & \vdots\\ g_{n-1}^0 & g_{n-1}^1 & \cdots & g_{n-1}^{n-1}\\ \end{pmatrix}. \]
That is:
\[ f_{n-1}(x,y) = \sum_{\ell=0}^{n-1}\sum_{m=0}^{n-1} g_\ell^m T_\ell(2x-1) T_m\left(\frac{2y}{1-x}-1\right), \]
and ft_execute_cheb2tri converts them back.
Proriol polynomials on the simplex with vertices \((0,0,0)\), \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\) are:
\[ \tilde{P}_{k,\ell,m}^{(\alpha,\beta,\gamma,\delta)}(x,y,z) = (2-2x)^{\ell+m} \tilde{P}_{k-\ell-m}^{(2\ell+2m+\beta+\gamma+\delta+2,\alpha)}(2x-1) \left(2-\frac{2y}{1-x}\right)^m \tilde{P}_\ell^{(2m+\gamma+\delta+1,\beta)}\left(\frac{2y}{1-x}-1\right) \tilde{P}_m^{(\delta,\gamma)}\left(\frac{2z}{1-x-y}-1\right), \]
where the tilde implies that the univariate Jacobi polynomials are orthonormal. A degree- \((n-1)\) expansion in Proriol polynomials is given by:
\[ f_{n-1}(x,y,z) = \sum_{k=0}^{n-1}\sum_{\ell=0}^k\sum_{m=0}^{k-\ell} f_{k,\ell}^m \tilde{P}_{k,\ell,m}^{(\alpha,\beta,\gamma,\delta)}(x,y,z) = \sum_{m=0}^{n-1}\sum_{\ell=0}^{n-1-m}\sum_{k=\ell+m}^{n-1} f_{k,\ell}^m \tilde{P}_{k,\ell,m}^{(\alpha,\beta,\gamma,\delta)}(x,y,z). \]
If Proriol expansion coefficients are organized into the rank- \(3\) array whose \(m^{\rm th}\) slice is:
\[ F[0:n-m-1,0:n-m-1,m] = \begin{pmatrix} f_{m,0}^m & f_{1+m,1}^m & f_{2+m,2}^m & \cdots & f_{n-1,n-1-m}^m\\ \vdots & \vdots & \vdots & \ddots & 0\\ f_{n-3,0}^m & f_{n-2,1}^m & f_{n-1,2}^m & & \vdots\\ f_{n-2,0}^m & f_{n-1,1}^m & 0 & \cdots & 0\\ f_{n-1,0}^m & 0 & 0 & \cdots & 0\\ \end{pmatrix}, \]
then ft_plan_tet2cheb creates the appropriate ft_tetrahedral_harmonic_plan, and ft_execute_tet2cheb returns the trivariate Chebyshev coefficients stored in the rank- \(3\) array \(G\). That is:
\[ f_{n-1}(x,y,z) = \sum_{k=0}^{n-1}\sum_{\ell=0}^{n-1}\sum_{m=0}^{n-1} g_{k,\ell}^m T_k(2x-1) T_\ell\left(\frac{2y}{1-x}-1\right) T_m\left(\frac{2z}{1-x-y}-1\right). \]
Complex-valued orthonormal spin-weighted spherical harmonics in \(L^2(\mathbb{S}^2)\) are:
\begin{align*} Y_{\ell,m}^s(\theta,\varphi) & = \frac{e^{\ii m\varphi}}{\sqrt{2\pi}} \sqrt{(\ell+\tfrac{1}{2})\frac{(\ell+\ell_0)!(\ell-\ell_0)!}{(\ell+\ell_1)!(\ell-\ell_1)!}} \sin^{\abs{m+s}}(\tfrac{\theta}{2})\cos^{\abs{m-s}}(\tfrac{\theta}{2}) P_{\ell-\ell_0}^{(\abs{m+s},\abs{m-s})}(\cos\theta), \end{align*}
where \(P_n^{(\alpha,\beta)}(\cos\theta)\) are the Jacobi polynomials and \(\ell_0 = \max\{\abs{m},\abs{s}\}\) and \(\ell_1 = \min\{\abs{m},\abs{s}\}\). A degree- \((n-1)\) expansion in spin-weighted spherical harmonics is given by:
\[ f_{n-1}^s(\theta,\varphi) = \sum_{\ell=\ell_0}^{n-1}\sum_{m=-\ell}^{+\ell} f_\ell^m Y_{\ell,m}^s(\theta,\varphi). \]
If spin-weighted spherical harmonic expansion coefficients with \(s=2\), for example, are organized into the array:
\[ F = \begin{pmatrix} f_2^0 & f_2^{-1} & f_2^1 & f_2^{-2} & f_2^2 & \cdots & f_{n-1}^{1-n} & f_{n-1}^{n-1}\\ f_3^0 & f_3^{-1} & f_3^1 & f_3^{-2} & f_3^2 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ f_{n-1}^0 & f_{n-1}^{-1} & f_{n-1}^1 & f_{n-1}^{-2} & f_{n-1}^2 & & \vdots & \vdots\\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\ \end{pmatrix}, \]
then ft_execute_spinsph2fourier returns the bivariate Fourier coefficients:
\[ G = \begin{pmatrix} g_0^0 & g_0^{-1} & g_0^1 & \cdots & g_0^{1-n} & g_0^{n-1}\\ g_1^0 & g_1^{-1} & g_1^1 & \cdots & g_1^{1-n} & g_1^{n-1}\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ g_{n-2}^0 & g_{n-2}^{-1} & g_{n-2}^1& \cdots & g_{n-2}^{1-n} & g_{n-2}^{n-1}\\ g_{n-1}^0 & 0 & 0 & \cdots & g_{n-1}^{1-n} & g_{n-1}^{n-1}\\ \end{pmatrix}. \]
That is:
\[ f_{n-1}^s(\theta,\varphi) = \sum_{\ell=0}^{n-1}\sum_{m=1-n}^{n-1} g_\ell^m \frac{e^{\ii m\varphi}}{\sqrt{2\pi}} \left\{\begin{array}{ccc} \cos(\ell\theta) & {\rm for} & m+s~{\rm even},\\ \sin((\ell+1)\theta) & {\rm for} & m+s~{\rm odd}.\end{array}\right. \]
This convention has positive real phase, which implies identities such as:
\[ \overline{Y}_{\ell,m}^s(\theta,\varphi) = Y_{\ell,-m}^{-s}(\theta,\varphi). \]
Spherical Gauss–Laguerre functions are:
\[ \psi_{k,\ell,m}(r,\theta,\varphi) = r^\ell \tilde{L}_{k-\ell}^{(2\ell+2)}(r) Y_\ell^m(\theta,\varphi), \]
where the \(Y_\ell^m(\theta,\varphi)\) are spherical harmonics, and the \(\tilde{L}_n^{(\alpha)}(r)\) are normalized generalized Laguerre polynomials. A degree- \((n-1)\) expansion in spherical Gauss–Laguerre functions is given by:
\[ f_{n-1}(r,\theta,\varphi) = \sum_{k=0}^{n-1}\sum_{\ell=0}^k\sum_{m=-\ell}^{+\ell}f_{k,\ell}^m \psi_{k,\ell,m}(r,\theta,\varphi). \]
1.9.5