Appendix 1: Evaluation of Covariance Matrices
In this section we compute the covariance matrix \(\Sigma _{\ell }(x,y)\) for the 10-dimensional random vector \(Z_{\ell ;x,y}\), which combines the gradient and the elements of the Hessian evaluated at x, y. \(\Sigma _{\ell }(x,y)\) depends only on the geodesic distance \(\phi =d(x,y)\), so, abusing notation, we shall write \(\Sigma _{\ell }(x,y)=\Sigma _{\ell }(\phi )\) whenever convenient, and similarly for the other functions we shall deal with. The computations are quite lengthy, but they do not require sophisticated arguments, other than iterative derivations of Legendre polynomials. It is convenient to write these matrices in block-diagonal form, i.e.,
$$\begin{aligned} \Sigma _{\ell }(\phi )=\left( \begin{array}{c@{\quad }c} A_{\ell }(\phi ) &{} B_{\ell }(\phi ) \\ B^{t}_{\ell }(\phi ) &{} C_{\ell }(\phi ) \end{array} \right) . \end{aligned}$$
In particular, the \(A_\ell \) component collects the variances of the gradient terms, and it is given by
$$\begin{aligned} A_{\ell }(x,y)_{4\times 4}&=\left. \mathbb {E}\left[ \left( \begin{array}{c} \nabla f_{\ell }(\bar{x})^{t} \\ \nabla f_{\ell }(\bar{y})^{t} \end{array} \right) \left( \begin{array}{c@{\quad }c} \nabla f_{\ell }(x)&\nabla f_{\ell }(y) \end{array} \right) \right] \right| _{{x=\bar{x}},{y=\bar{y}}}\\&=\left( \begin{array}{c@{\quad }c} a_{\ell }(x,x) &{} a_{\ell }(x,y) \\ a_{\ell }(y,x) &{} a_{\ell }(y,y) \end{array} \right) , \end{aligned}$$
where
$$\begin{aligned} a_{\ell }(x,x)= & {} \left. \left( \begin{array}{c@{\quad }c} e_{1}^{\bar{x}}e_{1}^{x}r_{\ell }(\bar{x},x) &{} e_{1}^{\bar{x} }e_{2}^{x}r_{\ell }(\bar{x},x) \\ e_{2}^{\bar{x}}e_{1}^{x}r_{\ell }(\bar{x},x) &{} e_{2}^{\bar{x} }e_{2}^{x}r_{\ell }(\bar{x},x) \end{array} \right) \right| _{x=\bar{x}}, \\ a_{\ell }(x,y)= & {} \left. \left( \begin{array}{c@{\quad }c} e_{1}^{\bar{x}}e_{1}^{y}r_{\ell }(\bar{x},y) &{} e_{1}^{\bar{x} }e_{2}^{y}r_{\ell }(\bar{x},y) \\ e_{2}^{\bar{x}}e_{1}^{y}r_{\ell }(\bar{x},y) &{} e_{2}^{\bar{x} }e_{2}^{y}r_{\ell }(\bar{x},y) \end{array} \right) \right| _{x=\bar{x}}, \end{aligned}$$
and
$$\begin{aligned}r_{\ell }(x,y)=\mathbb {E}[f_{\ell }(x) f_{\ell }(y)]=P_{\ell }(\cos d(x,y)), \end{aligned}$$
with \(h(x,y)=\cos d(x,y)=\cos \theta _x \cos \theta _y+\sin \theta _x \sin \theta _y \cos (\varphi _x-\varphi _y)\). Then, for example, computing the derivatives explicitly, we have
$$\begin{aligned} e_{1}^{\bar{x}}e_{1}^{x}r_{\ell }(\bar{x},x)=P''_{\ell }(h(\bar{x},x)) \frac{\partial }{\partial \theta _{\bar{x}}} h(\bar{x},x) \frac{\partial }{\partial \theta _{x}} h(\bar{x},x)+P'_{\ell }(h(\bar{x},x)) \frac{\partial }{\partial \theta _{\bar{x}}} \frac{\partial }{\partial \theta _{x}} h(\bar{x},x), \end{aligned}$$
where
$$\begin{aligned} \frac{\partial }{\partial \theta _{\bar{x}}} h(\bar{x},x)&=\left. -\cos \theta _{{x}} \sin \theta _{\bar{x}}+\cos \theta _{\bar{x}} \sin \theta _{{x}} \cos ( \varphi _{x}-\varphi _{\bar{x}}) \right| _{x=\bar{x}=(\pi /2,\varphi _{x})}=0,\\ \frac{\partial }{\partial \theta _{x}} h(\bar{x},x)&=\left. -\cos \theta _{\bar{x}} \sin \theta _{{x}}+\cos \theta _{{x}} \sin \theta _{\bar{x}} \cos ( \varphi _{x}-\varphi _{\bar{x}}) \right| _{x=\bar{x}=(\pi /2,\varphi _{x})}=0,\\ \frac{\partial }{\partial \theta _{\bar{x}}} \frac{\partial }{\partial \theta _{x}} h(\bar{x},x)&=\left. \sin \theta _{\bar{x}} \sin \theta _{{x}}+\cos \theta _{{x}} \cos \theta _{\bar{x}} \cos ( \varphi _{x}-\varphi _{\bar{x}}) \right| _{x=\bar{x}=(\pi /2,\varphi _{x})}=1. \end{aligned}$$
We then write
$$\begin{aligned} \left. a_{\ell }(x,x)\right| _{x=(\pi /2,\varphi _{x})}=\left. a_{\ell }(y,y)\right| _{y=(\pi /2,0)}=\left( \begin{array}{c@{\quad }c} P_{\ell }^{\prime }(1) &{} 0 \\ 0 &{} P_{\ell }^{\prime }(1) \end{array} \right) , \end{aligned}$$
and, again with some slight abuse of notation,
$$\begin{aligned} \left. a_{\ell }(x,y)\right| _{{x=(\pi /2,\varphi _{x})},{y=(\pi /2,0)} }&=\left. a_{\ell }(y,x)\right| _{{x=(\pi /2,\varphi _{x})},{ y=(\pi /2,0)}}\\&=\left( \begin{array}{c@{\quad }c} \alpha _{1,\ell }(\phi ) &{} 0 \\ 0 &{} \alpha _{2,\ell }(\phi ) \end{array} \right) , \end{aligned}$$
where as we recalled before \(\phi =d(x,y)\) and
$$\begin{aligned} \alpha _{1,\ell }(\phi )= & {} P_{\ell }^{\prime }(\cos \phi ),\\ \alpha _{2,\ell }(\phi )= & {} -\sin ^{2}\phi P_{\ell }^{\prime \prime }(\cos \phi )+\cos \phi P_{\ell }^{\prime }(\cos \phi ). \end{aligned}$$
Now recall that \(P_{\ell }^{\prime }(1)=\frac{\ell (\ell +1)}{2}\), for \( \lambda _{\ell }=\ell (\ell +1)\); hence we have
$$\begin{aligned} A_{\ell }(\phi )=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \frac{\lambda _{\ell }}{2} &{} 0 &{} \alpha _{1,\ell }(\phi ) &{} 0 \\ 0 &{} \frac{\lambda _{\ell }}{2} &{} 0 &{} \alpha _{2,\ell }(\phi ) \\ \alpha _{1,\ell }(\phi ) &{} 0 &{} \frac{\lambda _{\ell }}{2} &{} 0 \\ 0 &{} \alpha _{2,\ell }(\phi ) &{} 0 &{} \frac{\lambda _{\ell }}{2} \end{array} \right) . \end{aligned}$$
The matrix \(B_{\ell }\) collects the covariances between first- and second-order derivatives, and is given by
$$\begin{aligned} B_{\ell }(x,y)_{4\times 6}&=\left. \mathbb {E}\left[ \left( \begin{array}{c} \nabla f_{\ell }(\bar{x})^{t} \\ \nabla f_{\ell }(\bar{y})^{t} \end{array} \right) \left( \begin{array}{c@{\quad }c} \nabla ^{2}f_{\ell }(x)&\nabla ^{2}f_{\ell }(y) \end{array} \right) \right] \right| _{{x=\bar{x}},{y=\bar{y}}}\\&=\left( \begin{array}{c@{\quad }c} b_{\ell }(x,x) &{} b_{\ell }(x,y) \\ b_{\ell }(y,x) &{} b_{\ell }(y,y) \end{array} \right) . \end{aligned}$$
It is well known that for Gaussian isotropic processes, for \(i,j=1,2\), the second derivatives \(e^x_i e^x_j f_\ell (x)\) are independent of \(e_i^x f_\ell (x)\) at every fixed point \(x \in \mathcal{S}^2\); see, e.g., [2, Sect. 5.5]. We then have
$$\begin{aligned} \left. b_{\ell }(x,x)\right| _{x=(\pi /2,\varphi _{x})}=\left. b_{\ell }(y,y)\right| _{y=(\pi /2,0)}=\left( \begin{array}{c@{\quad }c@{\quad }c} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{array} \right) , \end{aligned}$$
while
$$\begin{aligned} \left. b_{\ell }(x,y)\right| _{{x=(\pi /2,\varphi _{x})},{y=(\pi /2,0)} }&=\left( \begin{array}{c@{\quad }c@{\quad }c} 0 &{} \beta _{1,\ell }(\phi ) &{} 0 \\ \beta _{2,\ell }(\phi ) &{} 0 &{} \beta _{3,\ell }(\phi ) \end{array} \right) \\&=-\left. b_{\ell }(y,x)\right| _{{x=(\pi /2,\varphi _{x})},{y=(\pi /2,0)}}. \end{aligned}$$
Here we have introduced the functions
$$\begin{aligned} \beta _{1,\ell }(\phi )= & {} \sin \phi P_{\ell }^{\prime \prime }(\cos \phi ),\\ \beta _{2,\ell }(\phi )= & {} \sin \phi \cos \phi P_{\ell }^{\prime \prime }(\cos \phi )+\sin \phi P_{\ell }^{\prime }(\cos \phi ),\\ \beta _{3,\ell }(\phi )= & {} -\sin ^{3}\phi P_{\ell }^{\prime \prime \prime }(\cos \phi )+3\sin \phi \cos \phi P_{\ell }^{\prime \prime }(\cos \phi )+\sin \phi P_{\ell }^{\prime }(\cos \phi ).\end{aligned}$$
Finally, the matrix \(C_{\ell }\) contains the variances of second-order derivatives, and we have
$$\begin{aligned} C_{\ell }(x,y)_{6\times 6}&=\left. \mathbb {E}\left[ \begin{array}{c} \left( \begin{array}{c} \nabla ^{2}f_{\ell }(\bar{x})^{t} \\ \nabla ^{2}f_{\ell }(\bar{y})^{t} \end{array} \right) \left( \begin{array}{c@{\quad }c} \nabla ^{2}f_{\ell }({x})&\nabla ^{2}f_{\ell }(\bar{y}) \end{array} \right) \end{array} \right] \right| _{{x=\bar{x}},{y=\bar{y}}}\\&=\left( \begin{array}{c@{\quad }c} c_{\ell }(x,x) &{} c_{\ell }(x,y) \\ c_{\ell }(y,x) &{} c_{\ell }(y,y) \end{array} \right) . \end{aligned}$$
Direct calculations yield
$$\begin{aligned} \left. c_{\ell }(x,y)\right| _{{x=(\pi /2,\varphi _{x})},{y=(\pi /2,0)} }&=\left. c_{\ell }(y,x)\right| _{{x=(\pi /2,\varphi _{x})},{ y=(\pi /2,0)}}\\&=\left( \begin{array}{c@{\quad }c@{\quad }c} \gamma _{1,\ell }(\phi ) &{} 0 &{} \gamma _{3,\ell }(\phi ) \\ 0 &{} \gamma _{2,\ell }(\phi ) &{} 0 \\ \gamma _{3,\ell }(\phi ) &{} 0 &{} \gamma _{4,\ell }(\phi ) \end{array} \right) , \end{aligned}$$
with
$$\begin{aligned} \gamma _{1,\ell }(\phi )= & {} (2+\cos ^{2}\phi )P_{\ell }^{\prime \prime }(\cos \phi )+\cos \phi P_{\ell }^{\prime }(\cos \phi ),\\ \gamma _{2,\ell }(\phi )= & {} -\sin ^{2}\phi P_{\ell }^{\prime \prime \prime }(\cos \phi )+\cos \phi P_{\ell }^{\prime \prime }(\cos \phi ),\\ \gamma _{3,\ell }(\phi )= & {} -\sin ^{2}\phi \cos \phi P_{\ell }^{\prime \prime \prime }(\cos \phi )+\left( -2\sin ^{2}\phi +\cos ^{2}\phi \right) P_{\ell }^{\prime \prime }(\cos \phi )\\&+\cos \phi P_{\ell }^{\prime }(\cos \phi ),\\ \gamma _{4,\ell }(\phi )= & {} \sin ^{4}\phi P_{\ell }^{\prime \prime \prime \prime }(\cos \phi )-6\sin ^{2}\phi \cos \phi P_{\ell }^{\prime \prime \prime }(\cos \phi )\\&+\left( -4\sin ^{2}\phi +3\cos ^{2}\phi \right) P_{\ell }^{\prime \prime }(\cos \phi )+\cos \phi P_{\ell }^{\prime }(\cos \phi ). \end{aligned}$$
Since \(P_{\ell }^{\prime \prime }(1)=\frac{\lambda _\ell }{8}(\lambda _\ell -2)\), it immediately follows that
$$\begin{aligned} \left. c_{\ell }(x,x)\right| _{x=(\pi /2,\varphi _{x})}&=\left( \begin{array}{c@{\quad }c@{\quad }c} 3P_{\ell }^{\prime \prime }(1)+P_{\ell }^{\prime }(1) &{} 0 &{} P_{\ell }^{\prime \prime }(1)+P_{\ell }^{\prime }(1) \\ 0 &{} P_{\ell }^{\prime \prime }(1) &{} 0 \\ P_{\ell }^{\prime \prime }(1)+P_{\ell }^{\prime }(1) &{} 0 &{} 3P_{\ell }^{\prime \prime }(1)+P_{\ell }^{\prime }(1) \end{array} \right) \\&=\left( \begin{array}{c@{\quad }c@{\quad }c} \frac{\lambda _{\ell }}{8}[3\lambda _{\ell }-2] &{} 0 &{} \frac{\lambda _{\ell } }{8}[\lambda _{\ell }+2] \\ 0 &{} \frac{\lambda _{\ell }}{8}[\lambda _{\ell }-2] &{} 0 \\ \frac{\lambda _{\ell }}{8}[\lambda _{\ell }+2] &{} 0 &{} \frac{\lambda _{\ell }}{ 8}[3\lambda _{\ell }-2] \end{array} \right) \\&=\left. c_{\ell }(y,y)\right| _{y=(\pi /2,0)}. \end{aligned}$$
Appendix 2: The Conditional Covariance Matrix \(\Delta _{\ell }(\phi )\)
In this section we compute the conditional covariance matrices \(\Omega _{\ell }(\phi )\) and \(\Delta _{\ell }(\phi )\) (Eqs. (4.3), (4.4)). To simplify the notation we will write \(\alpha _{i}, \beta _{i}, \gamma _{i}\) for \(\alpha _{i,\ell }(\phi ),\beta _{i,\ell }(\phi )\) and \(\gamma _{i,\ell }(\phi )\); likewise we will adopt the shorthand notation \(A,B,C,\Omega , \Delta \) for \(A_{\ell }(\phi ), B_{\ell }(\phi ),C_{\ell }(\phi ),\Omega _{\ell }(\phi ) \) and \(\Delta _{\ell }(\phi )\), respectively.
Let us first compute explicitly the inverse matrix \(A^{-1}\); we write A as a block matrix
$$\begin{aligned} A=\left( \begin{array}{c@{\quad }c} a_1 &{} a_2 \\ a_2 &{} a_1 \end{array} \right) \end{aligned}$$
where
$$\begin{aligned} a_1=\left( \begin{array}{c@{\quad }c} \frac{\lambda _{\ell }}{2} &{} 0 \\ 0 &{} \frac{\lambda _{\ell }}{2} \end{array} \right) ,\quad a_2=\left( \begin{array}{c@{\quad }c} \alpha _{1} &{} 0 \\ 0 &{} \alpha _{2} \end{array} \right) , \end{aligned}$$
and we evaluate the following components:
$$\begin{aligned} (a_1-a_2a_1^{-1}a_2)^{-1}=\left( \begin{array}{c@{\quad }c} \frac{2\lambda _{\ell }}{\lambda _{\ell }^{2}-4\alpha _{1}^{2}} &{} 0 \\ 0 &{} \frac{2\lambda _{\ell }}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} \end{array} \right) , \end{aligned}$$
and
$$\begin{aligned} a_1^{-1}a_2=a_2a_1^{-1}=\left( \begin{array}{c@{\quad }c} \frac{2\alpha _{1}}{\lambda _{\ell }} &{} 0 \\ 0 &{} \frac{2\alpha _{2}}{\lambda _{\ell }} \end{array} \right) . \end{aligned}$$
Now, to invert blockwise A, we need to compute the main diagonal blocks
$$\begin{aligned} a_1^{-1}+a_1^{-1}a_2\left( a_1-a_2a_1^{-1}a_2\right) ^{-1}a_2a_1^{-1} =\left( a_1-a_2a_1^{-1}a_2\right) ^{-1}, \end{aligned}$$
and the off-diagonal blocks
$$\begin{aligned} -a_1^{-1}a_2\left( a_1-a_2a_1^{-1}a_2\right) ^{-1}= -\left( a_1-a_2a_1^{-1}a_2\right) ^{-1}a_2a_1^{-1}=-\left( \begin{array}{c@{\quad }c} \frac{4\alpha _{1}}{\lambda _{\ell }^{2}-4\alpha _{1}^{2}} &{} 0 \\ 0 &{} \frac{4\alpha _{2}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} \end{array} \right) . \end{aligned}$$
We then have
$$\begin{aligned} A^{-1}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \frac{2\lambda _{\ell }}{\lambda _{\ell }^{2}-4\alpha _{1}^{2}} &{} 0 &{} -\frac{ 4\alpha _{1}}{\lambda _{\ell }^{2}-4\alpha _{1}^{2}} &{} 0 \\ 0 &{} \frac{2\lambda _{\ell }}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} &{} 0 &{} - \frac{4\alpha _{2}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} \\ -\frac{4\alpha _{1}}{\lambda _{\ell }^{2}-4\alpha _{1}^{2}} &{} 0 &{} \frac{ 2\lambda _{\ell }}{\lambda _{\ell }^{2}-4\alpha _{1}^{2}} &{} 0 \\ 0 &{} -\frac{4\alpha _{2}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} &{} 0 &{} \frac{ 2\lambda _{\ell }}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} \end{array} \right) . \end{aligned}$$
We are now in position to compute the matrix \(B^{t}A^{-1}B\); indeed, we get:
$$\begin{aligned} B^{t}A^{-1}B= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0 &{} 0 &{} 0 &{} -\beta _{2} \\ 0 &{} 0 &{} -\beta _{1} &{} 0 \\ 0 &{} 0 &{} 0 &{} -\beta _{3} \\ 0 &{} \beta _{2} &{} 0 &{} 0 \\ \beta _{1} &{} 0 &{} 0 &{} 0 \\ 0 &{} \beta _{3} &{} 0 &{} 0 \end{array} \right) A^{-1}\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0 &{} 0 &{} 0 &{} 0 &{} \beta _{1} &{} 0 \\ 0 &{} 0 &{} 0 &{} \beta _{2} &{} 0 &{} \beta _{3} \\ 0 &{} -\beta _{1} &{} 0 &{} 0 &{} 0 &{} 0 \\ -\beta _{2} &{} 0 &{} -\beta _{3} &{} 0 &{} 0 &{} 0 \end{array} \right) \\= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \frac{2\lambda _{\ell }\beta _{2}^{2}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} &{} 0 &{} \frac{2\lambda _{\ell }\beta _{2}\beta _{3}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} &{} \frac{4\alpha _{2}\beta _{2}^{2}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} &{} 0 &{} \frac{4\alpha _{2}\beta _{2}\beta _{3}}{ \lambda _{\ell }^{2}-4\alpha _{2}^{2}} \\ 0 &{} \frac{2\lambda _{\ell }\beta _{1}^{2}}{\lambda _{\ell }^{2}-4\alpha _{1}^{2}} &{} 0 &{} 0 &{} \frac{4\alpha _{1}\beta _{1}^{2}}{\lambda _{\ell }^{2}-4\alpha _{1}^{2}} &{} 0 \\ \frac{2\lambda _{\ell }\beta _{2}\beta _{3}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} &{} 0 &{} \frac{2\lambda _{\ell }\beta _{3}^{2}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} &{} \frac{4\alpha _{2}\beta _{2}\beta _{3}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} &{} 0 &{} \frac{4\alpha _{2}\beta _{3}^{2}}{ \lambda _{\ell }^{2}-4\alpha _{2}^{2}} \\ \frac{4\alpha _{2}\beta _{2}^{2}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} &{} 0 &{} \frac{4\alpha _{2}\beta _{2}\beta _{3}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} &{} \frac{2\lambda _{\ell }\beta _{2}^{2}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} &{} 0 &{} \frac{2\lambda _{\ell }\beta _{2}\beta _{3}}{ \lambda _{\ell }^{2}-4\alpha _{2}^{2}} \\ 0 &{} \frac{4\alpha _{1}\beta _{1}^{2}}{\lambda _{\ell }^{2}-4\alpha _{1}^{2}} &{} 0 &{} 0 &{} \frac{2\lambda _{\ell }\beta _{1}^{2}}{\lambda _{\ell }^{2}-4\alpha _{1}^{2}} &{} 0 \\ \frac{4\alpha _{2}\beta _{2}\beta _{3}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2} } &{} 0 &{} \frac{4\alpha _{2}\beta _{3}^{2}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} &{} \frac{2\lambda _{\ell }\beta _{2}\beta _{3}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} &{} 0 &{} \frac{2\lambda _{\ell }\beta _{3}^{2}}{\lambda _{\ell }^{2}-4\alpha _{2}^{2}} \end{array} \right) , \end{aligned}$$
From Appendix 1 we have
$$\begin{aligned} C=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 3\frac{\lambda _{\ell }(\lambda _{\ell }-2)}{8}+\frac{\lambda _{\ell }}{2} &{} 0 &{} \frac{\lambda _{\ell }(\lambda _{\ell }-2)}{8}+\frac{\lambda _{\ell }}{2} &{} \gamma _{1} &{} 0 &{} \gamma _{3} \\ 0 &{} \frac{\lambda _{\ell }(\lambda _{\ell }-2)}{8} &{} 0 &{} 0 &{} \gamma _{2} &{} 0\\ \frac{\lambda _{\ell }(\lambda _{\ell }-2)}{8}+\frac{\lambda _{\ell }}{2} &{} 0 &{} 3\frac{\lambda _{\ell }(\lambda _{\ell }-2)}{8}+\frac{\lambda _{\ell }}{2} &{} \gamma _{3} &{} 0 &{} \gamma _{4} \\ \gamma _{1} &{} 0 &{} \gamma _{3} &{} 3\frac{\lambda _{\ell }(\lambda _{\ell }-2)}{ 8}+\frac{\lambda _{\ell }}{2} &{} 0 &{} \frac{\lambda _{\ell }(\lambda _{\ell }-2)}{8}+\frac{\lambda _{\ell }}{2} \\ 0 &{} \gamma _{2} &{} 0 &{} 0 &{} \frac{\lambda _{\ell }(\lambda _{\ell }-2)}{8} &{} 0\\ \gamma _{3} &{} 0 &{} \gamma _{4} &{} \frac{\lambda _{\ell }(\lambda _{\ell }-2)}{8 }+\frac{\lambda _{\ell }}{2} &{} 0 &{} 3\frac{\lambda _{\ell }(\lambda _{\ell }-2)}{8}+\frac{\lambda _{\ell }}{2} \end{array} \right) ; \end{aligned}$$
The remaining computations to obtain \(\Omega \) and \(\Delta \) are straightforward.
Appendix 3: Some Estimates on Legendre Polynomials
Let us first recall the following:
Lemma 9.1
(Hilb’s asymptotics, [25, p. 195, Theorem 8.21.6]) For any \(\varepsilon >0\) and any constant \(C>0\), we have
$$\begin{aligned} P_\ell (\cos \phi )=\left( \frac{\phi }{\sin \phi }\right) ^{1/2} J_0((\ell +1/2) \phi )+\delta _\ell (\phi ), \end{aligned}$$
where \(J_\nu \) is the Bessel function of the first kind, \(P_\ell \) denotes Legendre polynomials, and the error term satisfies
$$\begin{aligned} \delta _\ell (\phi ) \ll {\left\{ \begin{array}{ll} \phi ^2 O(1), &{} 0<\phi <C/\ell , \\ \phi ^{1/2} O(\ell ^{-3/2}), &{} C/ \ell \le \phi , \end{array}\right. } \end{aligned}$$
uniformly w.r.t. \(\ell \ge 1\) and \(\phi \in [0, \pi -\varepsilon ]\).
Lemma 9.2
The following asymptotic representation for the Bessel functions of the first kind holds:
$$\begin{aligned} J_0(x)&=\left( \frac{2}{\pi x} \right) ^{1/2} \cos (x- \pi /4) \sum _{k=0}^\infty (-1)^k g(2k) \; (2 x)^{-2 k} \\&\quad +\left( \frac{2}{\pi x} \right) ^{1/2} \cos (x+ \pi /4) \sum _{k=0}^\infty (-1)^k g(2k+1)\; (2 x)^{-2 k-1}, \end{aligned}$$
where \(\varepsilon >0,|\arg x|\le \pi -\varepsilon ,g(0)=1\) and \(g(k)=\frac{(-1)(-3^2) \cdots (-(2k-1)^2)}{2^{2k} k!}=(-1)^k \frac{[(2k)!!]^2}{2^{2k} k!}\).
For a proof of Lemma 9.2, see [19, Sect. 5.11]. In the rest of the paper we use the following notation:
Lemma 9.3
For any constant \(C>0\), we have, uniformly for \(\ell \ge 1\) and \(\phi \in [C/\ell ,\pi /2]\):
$$\begin{aligned} P_{\ell }^{\prime }(\cos \phi )= & {} \sqrt{\frac{2}{\pi }}\frac{\ell ^{1-1/2}}{ \sin ^{1+1/2}\phi }\left[ \sin \psi _{\ell }^{-}-\frac{1}{8\ell \phi }\sin \psi _{\ell }^{+}\right] +R_{1}(\ell ,\phi ), \end{aligned}$$
(9.1)
$$\begin{aligned} P_{\ell }^{\prime \prime }(\cos \phi )= & {} \sqrt{\frac{2}{\pi }}\frac{\ell ^{2-1/2}}{\sin ^{2+1/2}\phi }\left[ -\cos \psi _{\ell }^{-}+\frac{1}{8\ell \phi }\cos \psi _{\ell }^{+}\right] \nonumber \\&-\sqrt{\frac{2}{\pi }}\frac{\ell ^{1-1/2}}{\sin ^{3+1/2}\phi }\left[ \cos \psi _{\ell -1}^{+}+\frac{1}{8\ell \phi }\cos \psi _{\ell -1}^{-}\right] +R_{2}(\ell ,\phi ), \end{aligned}$$
(9.2)
$$\begin{aligned} P_{\ell }^{\prime \prime \prime }(\cos \phi )= & {} \sqrt{\frac{2}{\pi }}\frac{ \ell ^{3-1/2}}{\sin ^{3+1/2}\phi }\left[ \cos \psi _{\ell }^{+}+\frac{1}{8\ell \phi }\cos \psi _{\ell }^{-}\right] \nonumber \\&+\sqrt{\frac{2}{\pi }}\frac{\ell ^{2-1/2}}{\sin ^{4+1/2}\phi }\left[ \frac{1 }{2}\left( \cos \psi _{\ell +1}^{-}+5\cos \psi _{\ell -1}^{-}\right) \right. \nonumber \\&\left. -\frac{1}{8\ell \phi }\frac{1}{2}\left( \cos \psi _{\ell +1}^{+}+5\cos \psi _{\ell -1}^{+}\right) \right] \nonumber \\&-\sqrt{\frac{2}{\pi }}\frac{\ell ^{1-1/2}}{\sin ^{5+1/2}\phi }\left[ 3\cos \phi \sin \psi _{\ell -1}^{-}-\frac{1}{8\ell \phi }3\cos \phi \sin \psi _{\ell -1}^{+}\right] \nonumber \\&+R_{3}(\ell ,\phi ), \end{aligned}$$
(9.3)
$$\begin{aligned} P_{\ell }^{\prime \prime \prime \prime }(\cos \phi )= & {} \sqrt{\frac{2}{\pi }} \frac{\ell ^{4-1/2}}{\sin ^{4+1/2}\phi }\left[ \cos \psi _{\ell }^{-}-\frac{1}{ 8\ell \phi }\cos \psi _{\ell }^{+}\right] \nonumber \\&+\sqrt{\frac{2}{\pi }}\frac{\ell ^{3-1/2}}{\sin ^{5+1/2}\phi }\left[ -\frac{3}{2} \left( \sin \psi _{\ell +1}^{-}+3\sin \psi _{\ell -1}^{-}\right) \right. \nonumber \\&\left. +\frac{1}{8\ell \phi } \frac{3}{2}(\sin \psi _{\ell +1}^{+}+3\sin \psi _{\ell -1}^{+})\right] \nonumber \\&+\sqrt{\frac{2}{\pi }}\frac{\ell ^{2-1/2}}{\sin ^{6+1/2}\phi }\left[ -\frac{1}{2} \left( \cos \psi _{\ell +2}^{-}+16\cos \psi _{\ell }^{-}+13\cos \psi _{\ell -2}^{-}\right) \right. \nonumber \\&\left. +\frac{1}{8\ell \phi }\frac{1}{2}(\cos \psi _{\ell +2}^{+}+16\cos \psi _{\ell }^{+}+13\cos \psi _{\ell -2}^{+})\right] \nonumber \\&+\sqrt{\frac{2}{\pi }}\frac{\ell ^{1-1/2}}{\sin ^{7+1/2}\phi }\left[ -3(5-4\sin ^{2}\phi )\cos \psi _{\ell -1}^{+}\right. \nonumber \\&\left. -\frac{1}{8\ell \phi }3(5-4\sin ^{2}\phi )\cos \psi _{\ell -1}^{-}\right] +R_{4}(\ell ,\phi ), \end{aligned}$$
(9.4)
where \(\psi _{\ell +k}^{\pm }=(\ell +k+1/2)\phi \pm \pi /4\).
Proof
By applying Hilb’s asymptotics in Lemma 9.1 and Lemma 9.2, we obtain
$$\begin{aligned} P_{\ell +\alpha }(\cos \phi )&= \cos \psi ^-_{\ell +\alpha } \sum _{k=0}^{\infty } h(2k) [s_{2k,r}(\ell ,\phi )+\sigma _{2k,r} (\ell ,\alpha ,\phi )]\\&\quad + \cos \psi ^+_{\ell +\alpha } \sum _{k=0}^{\infty } h(2k+1) [s_{2k+1,r}(\ell ,\phi )+\sigma _{2k+1,r} (\ell ,\alpha ,\phi )]\\&\quad +\phi ^{1/2} O((\ell +\alpha )^{-3/2}). \end{aligned}$$
where \(r=0,1,2 \ldots \) and
$$\begin{aligned} h(2k)&=\sqrt{\frac{2}{\pi }} (-1)^k g(2k) 2^{-2k}, \\ h(2k+1)&=\sqrt{\frac{2}{\pi }} (-1)^k g(2k+1) 2^{-2k-1},\\ s_{2k,r}(\ell ,\phi )&= \frac{\phi ^{-2k}}{\sqrt{\sin \phi }}\frac{1}{\ell ^{2k+1/2}} \sum _{n=0}^{r} \left( {\begin{array}{c}-2k-1/2\\ n\end{array}}\right) \left( \frac{1/2+\alpha }{\ell }\right) ^n,\\ s_{2k+1,r}(\ell ,\phi )&= \frac{\phi ^{-2k-1}}{\sqrt{\sin \phi }}\frac{1}{\ell ^{2k+3/2}} \sum _{n=0}^{r} \left( {\begin{array}{c}-2k-3/2\\ n\end{array}}\right) \left( \frac{1/2+\alpha }{\ell }\right) ^n,\\ \sigma _{2k,r} (\ell ,\alpha ,\phi )&=\frac{\phi ^{-2k}}{\sqrt{\sin \phi }}\frac{1}{\ell ^{2k+1/2}} \sum _{n=r+1}^{\infty } \left( {\begin{array}{c}-2k-1/2\\ n\end{array}}\right) \left( \frac{1/2+\alpha }{\ell }\right) ^n,\\ \sigma _{2k+1,r} (\ell ,\alpha ,\phi )&= \frac{\phi ^{-2k-1}}{\sqrt{\sin \phi }}\frac{1}{\ell ^{2k+3/2}} \sum _{n=r+1}^{\infty } \left( {\begin{array}{c}-2k-3/2\\ n\end{array}}\right) \left( \frac{1/2+\alpha }{\ell }\right) ^n. \end{aligned}$$
To obtain the asymptotic behavior of the first derivative in (9.1) we first note that
$$\begin{aligned} P'_\ell (\cos \phi )&= \frac{\ell +1}{\sin ^2 \phi } [\cos \phi P_\ell (\cos \phi )-P_{\ell +1}(\cos \phi )] \end{aligned}$$
where
$$\begin{aligned} \cos \phi P_\ell (\cos \phi )-P_{\ell +1}(\cos \phi )&= \cos \phi \cos \psi ^-_{\ell } \sum _{k=0}^\infty h(2k) [s_{2k,0}(\ell ,\phi )+\sigma _{2k,0}(\ell ,0,\phi )] \\&\quad + \cos \phi \cos \psi ^+_{\ell } \sum _{k=0}^\infty h(2k+1) [s_{2k+1,0}(\ell ,\phi )\\&\quad +\sigma _{2k+1,0}(\ell ,0,\phi )] + \cos \phi \; \phi ^{1/2} O(\ell ^{-3/2})\\&\quad - \cos \psi ^-_{\ell +1} \sum _{k=0}^\infty h(2k) [s_{2k,0}(\ell ,\phi )+\sigma _{2k,0}(\ell ,1,\phi )] \\&\quad - \cos \psi ^+_{\ell +1} \sum _{k=0}^\infty h(2k+1) [s_{2k+1,0}(\ell ,\phi )\\&\quad +\sigma _{2k+1,0}(\ell ,1,\phi )] + \; \phi ^{1/2} O\left( (\ell +1)^{-3/2}\right) . \end{aligned}$$
Now observe that
$$\begin{aligned} \cos \phi \cos \psi ^{\pm }_\ell -\cos \psi ^{\pm }_{\ell +1}&= \sin \phi \sin \psi ^{\pm }_\ell ; \end{aligned}$$
we obtain
$$\begin{aligned} \cos \phi P_\ell (\cos \phi )-P_{\ell +1}(\cos \phi )&= \sin \phi \sin \psi ^-_{\ell } h(0) s_{0,0}(\ell ,\phi ) \\&\quad +\sin \phi \sin \psi ^+_{\ell } h(1) s_{1,0}(\ell ,\phi ) +R'_1(\ell , \phi ), \end{aligned}$$
where
$$\begin{aligned} R'_1(\ell , \phi )=\ell ^{-3/2} \phi ^{-1/2}. \end{aligned}$$
Then (9.1) easily follows, since we get
$$\begin{aligned} P'_\ell (\cos \phi )&= \sqrt{\frac{2}{\pi }} \frac{\ell ^{1-1/2}}{\sin ^{1+1/2} \phi } \left[ \sin \psi ^-_\ell - \frac{1}{8 \ell \phi }\sin \psi ^+_\ell \right] \\&\quad +O\left( \ell ^{-1/2} \phi ^{-5/2}\right) . \end{aligned}$$
To prove the asymptotic behavior of the second derivative in (9.2) we start from
$$\begin{aligned} P''_\ell (\cos \phi )&=\frac{\ell +1}{\sin ^4 \phi } [ (1+2 \cos ^2 \phi +\ell \cos ^2 \phi ) P_\ell (\cos \phi )\nonumber \\&\quad -(5+2 \ell ) \cos \phi P_{\ell +1}(\cos \phi )+(\ell +2) P_{\ell +2}(\cos \phi ) ], \end{aligned}$$
(9.5)
and we note that
$$\begin{aligned}&\cos ^2 \phi \cos \psi ^{\pm }_\ell -2 \cos \phi \cos \psi ^{\pm }_{\ell +1} + \cos \psi ^{\pm }_{\ell +2} = -\sin ^2 \phi \cos \psi ^{\pm }_\ell , \\&(1+2 \cos ^2 \phi ) \cos \psi ^{\pm }_\ell - 5 \cos \phi \cos \psi ^{\pm }_{\ell +1} + 2 \cos \psi ^{\pm }_{\ell +2} =\pm \sin \phi \cos \psi ^{\mp }_{\ell -1}. \end{aligned}$$
Then we obtain
$$\begin{aligned}&(1+2 \cos ^2 \phi +\ell \cos ^2 \phi ) P_\ell (\cos \phi )-(5+2 \ell ) \cos \phi P_{\ell +1}(\cos \phi )\nonumber \\&\quad +(\ell +2) P_{\ell +2}(\cos \phi ) =-\ell \sin ^2 \phi \cos \psi ^-_{\ell } h(0) s_{0,0}(\ell ,\phi ) \nonumber \\&\quad - \ell \sin ^2 \phi \cos \psi ^+_{\ell } h(1) s_{1,0} (\ell ,\phi ) - \sin \phi \cos \psi ^+_{\ell -1} h(0) s_{0,0}(\ell ,\phi ) \nonumber \\&\quad + \sin \phi \cos \psi ^-_{\ell -1} h(1) s_{1,0} (\ell ,\phi ) + R'_2(\ell ,\phi ) \end{aligned}$$
(9.6)
where
$$\begin{aligned} R'_2(\ell ,\phi )= \ell ^{-3/2} \phi ^{-1/2}. \end{aligned}$$
Plugging (9.6) into (9.5), we finally have
$$\begin{aligned} P''_\ell (\cos \phi )&= \sqrt{\frac{2}{\pi } } \frac{ \ell ^{2-1/2}}{\sin ^{2+1/2} \phi } \left\{ - \cos \psi ^-_\ell + \frac{1}{8 \ell \phi } \cos \psi ^+_\ell \right\} \\&\quad +\sqrt{\frac{2}{\pi } } \frac{\ell ^{1-1/2}}{\sin ^{3+1/2} \phi } \left\{ - \cos \psi ^+_{\ell -1} -\frac{1}{8 \ell \phi } \cos \psi ^-_{\ell -1}\right\} \\&\quad +O\left( \ell ^{1/2} \phi ^{-9/2}\right) . \end{aligned}$$
To obtain the asymptotic behavior of the third derivative in (9.3), we write
$$\begin{aligned} P'''_\ell (x)&=\frac{\ell ^2(\ell +1)}{(x^2-1)^3} \left[ - x^3 P_\ell (x)+3 x^2 P_{\ell +1}(x)-3 x P_{\ell +2}(x)+ P_{\ell +3}(x)\right] \\&\quad +\frac{\ell (\ell +1)}{(x^2-1)^3} \left[ - \left( 3 x + 5 x^3\right) P_\ell (x)+ \left( 3+18 x^2\right) P_{\ell +1}(x)\right. \\&\left. \qquad - 18 xP_{\ell +2}(x)+5 P_{\ell +3}(x)\right] \\&\quad +\frac{\ell +1}{(x^2-1)^3} \left[ -\left( 9x+6 x^3\right) P_\ell (x)+\left( 6+27 x^2\right) P_{\ell +1}(x)\right. \\&\left. \qquad - 24 xP_{\ell +2}(x)+6 P_{\ell +3}(x)\right] . \end{aligned}$$
Now, for \(r=1\), we obtain, for example, that
$$\begin{aligned}&- \cos ^3 \phi P_\ell (\cos \phi )+3 \cos ^2 \phi P_{\ell +1}(\cos \phi )-3 \cos \phi P_{\ell +2}(\cos \phi )+ P_{\ell +3}(\cos \phi )\\&\quad = - \cos ^3 \phi \left[ \cos \psi ^-_{\ell } \sum _{k=0}^\infty \left[ h(2k) s_{2k,1}(\ell ,\phi ) + \sigma _{2k,1}(\ell ,0,\phi )\right] \right. \\&\left. \quad + \cos \psi ^+_{\ell } \sum _{k=0}^\infty \left[ h(2k+1) s_{2k+1,1}(\ell ,\phi ) + \sigma _{2k+1,1}(\ell ,0,\phi )\right] \right] \\&\quad +3 \cos ^2 \phi \left[ \cos \psi ^-_{\ell +1} \sum _{k=0}^\infty \left[ h(2k) s_{2k,1}(\ell ,\phi ) + \sigma _{2k,1}(\ell ,1,\phi ) \right] \right. \\&\left. \quad + \cos \psi ^+_{\ell +1} \sum _{k=0}^\infty \left[ h(2k+1) s_{2k+1,1}(\ell ,\phi ) + \sigma _{2k+1,1}(\ell ,1,\phi )\right] \right] \\&\quad -3 \cos \phi \left[ \cos \psi ^-_{\ell +2} \sum _{k=0}^\infty \left[ h(2k) s_{2k,1}(\ell ,\phi ) + \sigma _{2k,1}(\ell ,2,\phi )\right] \right. \\&\left. \quad + \cos \psi ^+_{\ell +2} \sum _{k=0}^\infty \left[ h(2k+1) s_{2k+1,1}(\ell ,\phi ) + \sigma _{2k+1,1}(\ell ,2,\phi )\right] \right] \\&\quad + \cos \psi ^-_{\ell +3} \sum _{k=0}^\infty \left[ h(2k) s_{2k,1}(\ell ,\phi ) + \sigma _{2k,1}(\ell ,3,\phi )\right] \\&\quad + \cos \psi ^+_{\ell +3} \sum _{k=0}^\infty \left[ h(2k+1) s_{2k+1,1}(\ell ,\phi ) + \sigma _{2k+1,1}(\ell ,3,\phi )\right] ; \end{aligned}$$
now exploiting the identities
$$\begin{aligned}&-\cos ^3 \phi \cos \psi ^{\pm }_{\ell }+3 \cos ^2 \phi \cos \psi ^{\pm }_{\ell +1}\\&\quad -3 \cos \phi \cos \psi ^{\pm }_{\ell +2}+\cos \psi ^{\pm }_{\ell +3}\\&= {\pm } \sin ^3 \phi \cos \psi _\ell ^{\mp },3 \cos ^2 \phi \cos \psi ^{\pm }_{\ell +1}-3\cdot 2 \cos \phi \cos \psi ^{\pm }_{\ell +2}\\&\quad + 3\cos \psi ^{\pm }_{\ell +3}= {\pm } 3 \sin ^2 \phi \cos \psi _{\ell +1}^{\mp }, \end{aligned}$$
we get
$$\begin{aligned}&- \cos ^3 \phi P_\ell (\cos \phi )+3 \cos ^2 \phi P_{\ell +1}(\cos \phi )-3 \cos \phi P_{\ell +2}(\cos \phi )+ P_{\ell +3}(\cos \phi )\\&\quad = - \sin ^3 \phi \cos \psi ^+_{\ell } h(0) s_{0,0}(\ell ,\phi ) +\sin ^3 \phi \cos \psi ^-_{\ell } h(1) s_{1,0}(\ell ,\phi )+R'_3(\ell ,\phi ). \end{aligned}$$
For the other terms we need to apply the following equalities:
$$\begin{aligned}&-\left( 3 \cos \phi +5 \cos ^3 \phi \right) \cos \psi ^{\pm }_{\ell }+(3+18 \cos ^2 \phi ) \cos \psi ^{\pm }_{\ell +1}\\&\quad -18 \cos \phi \cos \psi ^{\pm }_{\ell +2}+5 \cos \psi ^{\pm }_{\ell +3}\\&\quad =\frac{1}{2} \sin ^2 \phi \left[ \cos \psi ^{\pm }_{\ell +1}+5 \cos \psi ^{\pm }_{\ell -1}\right] ,\\&\quad \left( -9 \cos \phi -6 \cos ^3 \phi \right) \cos \psi ^{\pm }_\ell +(6+27 \cos ^2 \phi ) \cos \psi ^{\pm }_{\ell +1}\\&\quad -24 \cos \phi \cos \psi ^{\pm }_{\ell +2}+6 \cos \psi ^{\pm }_{\ell +3}=-3 \sin \phi \cos \phi \sin \psi ^{\pm }_{\ell -1}, \end{aligned}$$
and we get
$$\begin{aligned} P'''_\ell (\cos \phi )&=\sqrt{\frac{2}{\pi }}\frac{\ell ^{3-1/2}}{\sin ^{3+1/2} \phi } \left[ \cos \psi _{\ell }^+ + \frac{1}{8 \ell \phi } \cos \psi _{\ell }^- \right] \\&\quad - \sqrt{\frac{2}{\pi }} \frac{\ell ^{2-1/2} }{\sin ^{4+1/2} \phi } \left[ \frac{1}{2} \left( \cos \psi ^{-}_{\ell +1}+5 \cos \psi ^{-}_{\ell -1}\right) \right. \\&\left. \quad -\frac{1}{8 \ell \phi } \frac{1}{2} \left( \cos \psi ^{+}_{\ell +1}+5 \cos \psi ^{+}_{\ell -1}\right) \right] \\&\quad +\sqrt{\frac{2}{\pi }} \frac{\ell ^{1-1/2}}{\sin ^{5+1/2} \phi } \left[ 3 \cos \phi \sin \psi ^{-}_{\ell -1} - \frac{1}{8 \ell \phi } 3 \cos \phi \sin \psi ^{+}_{\ell -1} \right] \\&\quad +R_3(\ell ,\phi ) . \end{aligned}$$
Finally, to prove (9.4), we start from
$$\begin{aligned} P''''_\ell (x)&=\frac{\ell ^3(\ell +1)}{(x^2-1)^4} \left[ x^4 P_\ell (x)-4 x^3 P_{\ell +1}(x)+6 x^2 P_{\ell +2}(x)\right. \\&\left. \quad -4 x P_{\ell +3}(x)+P_{\ell +4}(x)\right] \\&+\frac{\ell ^2(\ell +1)}{(x^2-1)^4} \left[ \left( 9 x^4+6 x^2\right) P_\ell (x)-\left( 42 x^3+12 x\right) P_{\ell +1}(x)\right. \\&\left. \quad +\left( 66 x^2+6 \right) P_{\ell +2}(x) -42 x P_{\ell +3}(x)+9 P_{\ell +4}(x)\right] \\&+\frac{\ell (\ell +1)}{(x^2-1)^4} \left[ \left( 26 x^4+42 x^2+3\right) P_\ell (x)-\left( 146 x^3+78 x\right) P_{\ell +1}(x)\right. \\&\left. \quad +\left( 231 x^2+30 \right) P_{\ell +2}(x)-134 xP_{\ell +3}(x)+26 P_{\ell +4}(x)\right] \\&+\frac{\ell +1}{(x^2-1)^4} \left[ \left( 24 x^4+72 x^2+9\right) P_\ell (x)-\left( 168x^3+111 x\right) P_{\ell +1}(x)\right. \\&\left. \quad +\left( 246 x^2+36\right) P_{\ell +2}(x)-132 x P_{\ell +3}(x)+24 P_{\ell +4}(x)\right] . \end{aligned}$$
Let us introduce the further identities
$$\begin{aligned}&\cos ^4 \phi \cos \psi ^{\pm }_\ell -4 \cos ^3 \phi \cos \psi ^{\pm }_{\ell +1}+6 \cos ^2 \phi \cos \psi ^{\pm }_{\ell +2}\\&\qquad -4 \cos \phi \cos \psi ^{\pm }_{\ell +3}+\cos \psi ^{\pm }_{\ell +4}=\sin ^4 \phi \cos \psi ^{\pm }_\ell , \\&\cos ^4 \phi -4 \cos ^3 \phi +6 \cos ^2 \phi -4 \cos \phi +1=(\cos \phi -1)^4,\\&\left( 6 \cos ^2\phi +9 \cos ^4 \phi \right) \cos \psi ^{\pm }_{\ell }-\left( 12 \cos \phi +42\cos ^3\phi \right) \cos \psi ^{\pm }_{\ell +1}\\&\qquad +\left( 6+66 \cos ^2 \phi \right) \cos \psi ^{\pm }_{\ell +2} -42 \cos \phi \cos \psi ^{\pm }_{\ell +3} \\&\qquad +9 \cos \psi ^{\pm }_{\ell +4}=- \frac{3}{2} \sin ^3 \phi \left[ \sin \psi ^{\pm }_{\ell +1} +3 \sin \psi ^{\pm }_{\ell -1} \right] ,\\&\left( 6 \cos ^2\phi +9 \cos ^4 \phi \right) -\left( 12 \cos \phi +42\cos ^3\phi \right) +\left( 6+66 \cos ^2 \phi \right) \\&\quad \quad -42 \cos \phi +9 =3 (\cos \phi -1)^3 (3 \cos \phi -5),\\&\left( 3+42 \cos ^2 \phi +26 \cos ^4\phi \right) \cos \psi ^{\pm }_{\ell }+\left( -78 \cos \phi -146 \cos ^3 \phi \right) \\&\quad \cos \psi ^{\pm }_{\ell +1}+\left( 30+231 \cos ^2\phi \right) \cos \psi ^{\pm }_{\ell +2}-134 \cos \phi \cos \psi ^{\pm }_{\ell +3}+26 \cos \psi ^{\pm }_{\ell +4}\\&\quad =-\frac{1}{2} \sin ^2 \phi \left[ \cos \psi ^{\pm }_{\ell +2}+16 \cos \psi ^{\pm }_{\ell } +13 \cos \psi ^{\pm }_{\ell -2} \right] ,\\&\left( 3+42 \cos ^2 \phi +26 \cos ^4\phi \right) + \left( -78 \cos \phi -146 \cos ^3 \phi \right) \\&\qquad +\left( 30+231 \cos ^2\phi \right) -134 \cos \phi +26 \\&\quad =\left( \cos \phi -1\right) ^2 \left( 59-94 \cos \phi +26 \cos ^2 \phi \right) ,\\&\left( 9+72 \cos ^2 \phi +24 \cos ^4 \phi \right) \cos \psi ^{\pm }_{\ell }\\&\quad \quad +\left( -111 \cos \phi -168 \cos ^3 \phi \right) \cos \psi ^{\pm }_{\ell +1}+\left( 36+246 \cos ^2\phi \right) \cos \psi ^{\pm }_{\ell +2}\\&\quad -132 \cos \phi \cos \psi ^{\pm }_{\ell +3}+24 \cos \psi ^{\pm }_{\ell +4}={\pm }3 \sin \phi (5-4\sin ^2 \phi ) \cos \psi ^{\mp }_{\ell -1}, \end{aligned}$$
from which we obtain the dominant terms in (9.4). The analysis of the remainder terms is omitted for brevity’s sake. \(\square \)
In what follows we write \(f_{\ell }(\phi ) \simeq g_{\ell }(\phi )\), if there exists a constant \(c_0>0\) such that
$$\begin{aligned} -c_0 \; g_{\ell }(\phi ) \le f_{\ell }(\phi ) \le c_0 \; g_{\ell }(\phi ), \end{aligned}$$
for all \(\ell \ge 1\) and \(\phi \in [C/\ell , \pi /2]\). From Lemma 9.3 it follows immediately that:
Lemma 9.4
Uniformly in \(\ell \ge 1\) and \(\phi \in [C/\ell , \pi /2]\), we have
$$\begin{aligned}&P^{\prime }_\ell (\cos \phi ) \simeq \frac{\ell ^{1-1/2}}{ \sin ^{1+1/2} \phi } +R_1(\ell ,\phi ), \\&P^{\prime \prime }_\ell (\cos \phi ) \simeq \sum _{i=0}^1 \frac{ \ell ^{2-i-1/2}}{ \sin ^{2+i+1/2} \phi } +R_2(\ell ,\phi ),\\&P^{\prime \prime \prime }_\ell (\cos \phi ) \simeq \sum _{i=0}^2 \frac{ \ell ^{3-i-1/2}}{ \sin ^{3+i+1/2} \phi } +R_3(\ell ,\phi ),\\&P^{\prime \prime \prime \prime }_\ell (\cos \phi ) \simeq \sum _{i=0}^3 \frac{ \ell ^{4-i-1/2}}{ \sin ^{4+i+1/2} \phi }+R_4(\ell ,\phi ). \end{aligned}$$
Proof
From (9.1) and since \(\phi \in [C/\ell , \pi /2]\), we obtain
$$\begin{aligned} P^{\prime }_\ell (\cos \phi )&\le \sqrt{\frac{2}{\pi }} \frac{\ell ^{1-1/2}}{ \sin ^{1+1/2} \phi } \big [ 1+ \frac{1}{8 \ell \phi } \big ] \\&\quad +R_1(\ell ,\phi ) \le \sqrt{\frac{2}{\pi }} \frac{\ell ^{1-1/2}}{ \sin ^{1+1/2} \phi } \big [ 1+ \frac{1}{8 C} \big ] +R_1(\ell ,\phi ), \end{aligned}$$
and
$$\begin{aligned} P^{\prime }_\ell (\cos \phi )&\ge \sqrt{\frac{2}{\pi }} \frac{\ell ^{1-1/2}}{ \sin ^{1+1/2} \phi } \big [ -1 - \frac{1}{8 \ell \phi } \big ] +R_1(\ell ,\phi ) \\&\ge - \sqrt{\frac{2}{\pi }} \frac{\ell ^{1-1/2}}{ \sin ^{1+1/2} \phi } \big [ 1+ \frac{1}{8 C} \big ] +R_1(\ell ,\phi ). \end{aligned}$$
The proof is analogous in the other cases. \(\square \)
From Lemma 9.4 we obtain the following asymptotics for the elements of the covariance matrix \(\Sigma \).
Lemma 9.5
For every \(\ell \ge 1\) and \(\phi \in [C/\ell , \pi /2]\), we have
$$\begin{aligned} \frac{\alpha _{1,\ell }(\phi )}{\ell ^2}&\simeq \frac{1}{ \ell ^{1+1/2} \sin ^{1+1/2} \phi } + O(\ell ^{-5/2} \phi ^{-5/2}) \\ \frac{\alpha _{2,\ell }(\phi )}{\ell ^2}&\simeq \sum _{i=0}^1 \frac{ 1}{ \ell ^{i+1/2} \sin ^{i+1/2} \phi } + O(\ell ^{-3/2}\phi ^{-3/2}) \\ \frac{\beta _{1,\ell }(\phi )}{\ell ^3}&\simeq \sum _{i=0}^1 \frac{ 1}{ \ell ^{1+i+1/2} \sin ^{1+i+1/2} \phi } +O(\ell ^{-5/2}\phi ^{-5/2}) \\ \frac{\beta _{2,\ell }(\phi )}{\ell ^3}&\simeq \sum _{i=0}^1 \frac{1}{ \ell ^{1+i+1/2} \sin ^{1+i +1/2} \phi }+ \frac{1}{\ell ^{2+1/2} \sin ^{1/2} \phi } + O(\ell ^{-5/2} \phi ^{-5/2})\\ \frac{\beta _{3,\ell }(\phi )}{\ell ^3}&\simeq \sum _{i=0}^2 \frac{1}{ \ell ^{i+1/2} \sin ^{i+1/2} \phi } + \frac{1}{\ell ^{2+1/2} \sin ^{1/2} \phi } + O(\ell ^{-3/2} \phi ^{-3/2}) \\ \frac{\gamma _{1,\ell }(\phi )}{\ell ^4}&\simeq \sum _{i=0}^1 \frac{1}{ \ell ^{2+i+1/2} \sin ^{2+i+1/2} \phi }+ \frac{1}{\ell ^{3+1/2} \sin ^{1+1/2} \phi } +O\left( \ell ^{-7/2} \phi ^{-7/2}\right) \\ \frac{\gamma _{2,\ell }(\phi )}{\ell ^4}&\simeq \sum _{i=0}^1 \frac{1}{ \ell ^{1+i+1/2} \sin ^{1+i +1/2} \phi }+ \frac{1}{\ell ^{2+1/2} \sin ^{1/2} \phi } +O(\ell ^{-5/2} \phi ^{-5/2}) \\ \frac{\gamma _{3,\ell }(\phi )}{\ell ^4}&\simeq \sum _{i=0}^3 \frac{1}{ \ell ^{1+i+1/2} \sin ^{1+i+1/2} \phi } + \sum _{i=0}^3 \frac{1}{\ell ^{2+i+1/2} \sin ^{i+1/2} \phi } + O(\ell ^{-5/2} \phi ^{-5/2}) \\ \frac{\gamma _{4,\ell }(\phi )}{\ell ^4}&\simeq \sum _{i=0}^3 \frac{1}{ \ell ^{i+1/2} \sin ^{i+1/2} \phi }+ \sum _{i=0}^1 \frac{1}{\ell ^{2+i+1/2} \sin ^{i+1/2} \phi } + O\left( \ell ^{-3/2} \phi ^{-3/2}\right) . \end{aligned}$$
Lemma 9.6
For \(k=0,1,2,\ldots \) and \(n=1,2,3, \ldots \), we have
$$\begin{aligned} \frac{1}{\ell ^k} \int _{C/\ell }^{\pi /2} \frac{1}{\sin ^n \phi } \sin \phi d \phi = {\left\{ \begin{array}{ll} O(\ell ^{-k}) &{} \mathrm {for\;} n=1, \\ O(\ell ^{-k} \log \ell ) &{} \mathrm {for\;} n=2, \\ O(\ell ^{n-k-2}) &{} \mathrm {for\;} n\ge 3. \end{array}\right. } \end{aligned}$$
Lemma 9.7
For \(n=0,1,2, \ldots \), we have
$$\begin{aligned} \frac{1}{\ell ^{n+1/2}} \int _{C/\ell }^{\pi /2} \frac{1}{\sin ^{n+1/2} \phi } \sin \phi d \phi = {\left\{ \begin{array}{ll} O(\ell ^{-1/2}) &{} \mathrm {for\;} n=0, \\ O(\ell ^{-1-1/2}) &{} \mathrm {for\;} n=1, \\ O(\ell ^{-2}) &{} \mathrm {for\;} n\ge 2. \end{array}\right. } \end{aligned}$$
Lemma 9.8
For \(k,n=1,2,3, \ldots \), we have
$$\begin{aligned} \frac{1}{\ell ^k} \int _{C/\ell }^{\pi /2} \frac{1}{\sin ^n \phi } \frac{1}{\phi ^k} \sin \phi d \phi = {\left\{ \begin{array}{ll} O(\ell ^{-1} \log \ell ) &{} \mathrm {for\;} n+k=2, \\ O(\ell ^{n-2}) &{} \mathrm {for\;} n+k\ge 3. \end{array}\right. } \end{aligned}$$
Proof
We first note that
$$\begin{aligned} \frac{1}{\ell ^k} \int _{C/\ell }^{\pi /2} \frac{1}{\phi ^{n+k-1}} d \phi \le \frac{1}{\ell ^k} \int _{C/\ell }^{\pi /2} \frac{1}{\sin ^n \phi } \frac{1}{\phi ^k} \sin \phi d \phi \le \frac{1}{\ell ^k} \int _{C/\ell }^{\pi /2} \frac{1}{\sin ^{n+k-1} \phi } d \phi , \end{aligned}$$
then the conclusion follows from Lemma 9.6 and by observing that
$$\begin{aligned} \frac{1}{\ell ^k} \int _{C/\ell }^{\pi /2} \frac{1}{\phi ^{n+k-1}} d \phi = {\left\{ \begin{array}{ll} O(\ell ^{-1} \log \ell ) &{} \mathrm{for\;} n+k=2,\\ O(\ell ^{n-2})&{} \mathrm{for\;} n+k\ge 3. \end{array}\right. } \end{aligned}$$
\(\square \)
Appendix 4: Bounds for the Terms \(A_{0,\ell },A_{i,\ell }\) and \(A_{ij,\ell }\)
Proof of Lemma 4.2
By expanding the denominator in \(A_{0,\ell }\), we write
$$\begin{aligned} A_{0,\ell }&=\int _{C/\ell }^{\frac{\pi }{2}}\Big [1+2\frac{\alpha _{2,\ell }^{2}(\phi )}{\lambda _{\ell }^{2}}+O\Big (\frac{\alpha _{2,\ell }^{4}(\phi )}{\lambda _{\ell }^{4}}\Big )\Big ] \Big [1+2\frac{\alpha _{1,\ell }^{2}(\phi )}{\lambda _{\ell }^{2}}+O\Big (\frac{\alpha _{1,\ell }^{4}(\phi )}{\lambda _{\ell }^{4}}\Big )\Big ] \sin \phi d\phi \\&=\cos \left( \frac{C}{\ell } \right) +\int _{C/ \ell }^{\frac{\pi }{2}}\Big [ 2\frac{\alpha _{2,\ell }^{2}(\phi )}{\lambda _{\ell }^{2}}+O\Big (\frac{ \alpha _{2,\ell }^{4}(\phi )}{\lambda _{\ell }^{4}}\Big ) +2\frac{\alpha _{1,\ell }^{2}(\phi )}{\lambda _{\ell }^{2}}+O\Big (\frac{\alpha _{1,\ell }^{4}(\phi )}{\lambda _{\ell }^{4}} \Big )\\&\quad +\Big (2\frac{\alpha _{2,\ell }^{2}(\phi )}{\lambda _{\ell }^{2}}+O\Big (\frac{\alpha _{2,\ell }^{4}(\phi )}{\lambda _{\ell }^{4}} \Big )\Big ) \Big (2\frac{\alpha _{1,\ell }^{2}(\phi )}{\lambda _{\ell }^{2}}+O\Big (\frac{\alpha _{1,\ell }^{4}(\phi )}{\lambda _{\ell }^{4}}\Big ) \Big )\Big ]\sin \phi d\phi . \end{aligned}$$
The idea is that the leading term in this expansion produces the cancellation of the component \(\big ( \mathbb {E}[\mathcal{N}^c_I(f_{\ell })] \big )^2\). Indeed, in view of Remark 4.1, we have
$$\begin{aligned} 2 \lambda _{\ell }^{2} \cos \big ( \frac{C}{\ell } \big ) \; \iint _{I \times I} q(\mathbf{0}; t_1,t_2) d t_1 dt_2- \frac{\lambda _{\ell }^2}{4} \Big [ \int _I p_1^c(t) d t \Big ]^2&=O(\ell ^{2}). \end{aligned}$$
We now consider the rate of the terms
$$\begin{aligned}&2 \lambda _{\ell }^{2} \int _{C/\ell }^{\frac{\pi }{2}}\Big [ 2\frac{\alpha _{2,\ell }^{2}(\phi )}{\lambda _{\ell }^{2}}+2\frac{\alpha _{1,\ell }^{2}(\phi )}{\lambda _{\ell }^{2}}\\&\quad + 4\frac{\alpha _{2,\ell }^{2}(\phi )}{\lambda _{\ell }^{2}} \frac{\alpha _{1,\ell }^{2}(\phi )}{\lambda _{\ell }^{2}} +O\Big (\frac{\alpha _{1,\ell }^{4}(\phi )}{\lambda _{\ell }^{4}} \Big )\\&\quad +O\Big (\frac{\alpha _{2,\ell }^{4}(\phi )}{\lambda _{\ell }^{4}}\Big )\Big ] \sin \phi \; d\phi \; \iint _{I \times I}q(\mathbf{0}; t_1,t_2) d t_1 dt_2. \end{aligned}$$
We apply here Lemma 9.5 and Lemma 9.6 to identify the rate of the dominant term. In fact, from Lemma 9.5, we obtain the asymptotic behavior of each addend of the integrand function; then Lemma 9.6 gives the asymptotic behavior of their integrals. We immediately see that
$$\begin{aligned} \frac{1}{\lambda _{\ell }^{2}}\int _{C/\ell }^{ \frac{\pi }{2}} \alpha ^{2}_{2,\ell }(\phi ) \sin \phi d\phi =O(\ell ^{-1}). \end{aligned}$$
(10.1)
To obtain the multiplicative constant of the leading term (10.1) note that, from Lemma 9.3 and again applying Lemma 9.6, to determine the non-dominant terms, we can write
$$\begin{aligned} \frac{2}{\lambda _{\ell }^{2}}\int _{C/\ell }^{ \frac{\pi }{2}}\alpha ^2_{2,\ell }(\phi ) \sin \phi d\phi&=\frac{2}{\lambda _{\ell }^{2}}\int _{C/\ell }^{ \frac{\pi }{2}} \left[ -\sin ^{2}\phi P_{\ell }^{\prime \prime }(\cos \phi )+\cos \phi P_{\ell }^{\prime }(\cos \phi )\right] ^{2} \sin \phi d\phi \nonumber \\&=\frac{2}{\lambda _{\ell }^{2}}\int _{C/\ell }^{ \frac{\pi }{2}} \left\{ -\sin ^{2}\phi \sqrt{\frac{2}{\pi } } \frac{ \ell ^{2-1/2}}{ \sin ^{2+1/2} \phi } \right. \nonumber \\&\quad \left. \left[ - \cos \psi _\ell ^- + \frac{1}{8 \ell \phi } \cos \psi _\ell ^+ \right] \right\} ^{2} \sin \phi d\phi +O(\ell ^{-2} \log \ell )\nonumber \\&=\frac{4}{\pi } \int _{C/\ell }^{\frac{\pi }{2}} \frac{1}{ \ell \sin \phi } \left[ - \cos \psi _\ell ^- + \frac{1}{8 \ell \phi } \cos \psi _\ell ^+ \right] ^{2} \nonumber \\&\quad \sin \phi d\phi +O\left( \ell ^{-2} \log \ell \right) , \end{aligned}$$
(10.2)
where \(\psi _{\ell }^{\pm }=(\ell +1/2) \phi \pm \pi /4\). Applying Lemma 9.8 to get the asymptotic behavior of the integral (10.2), we have
$$\begin{aligned} \frac{2}{\lambda _{\ell }^{2}}\int _{C/\ell }^{ \frac{\pi }{2}}\alpha ^{2}_{2,\ell }(\phi ) \sin \phi d\phi&=\frac{4}{\pi } \frac{1}{\ell } \int _{C/\ell }^{\frac{\pi }{2}} \cos ^2 \psi _\ell ^- d\phi +O(\ell ^{-2} \log \ell )\\&= \frac{4}{\pi } \frac{1}{\ell } \int _{C/\ell }^{\frac{\pi }{2}} \left[ \frac{1}{2}+\frac{1}{2} \cos (2 \psi _{\ell }^{-})\right] d\phi +O(\ell ^{-2} \log \ell )\\&=\frac{2}{\pi } \frac{1}{\ell }\int _{C/\ell }^{\frac{\pi }{2}} d\phi +\frac{2}{\pi \ell }\int _{C/\ell }^{\frac{\pi }{2}} \cos (2\psi _{\ell }^{-}) d\phi +O(\ell ^{-2} \log \ell )\\&=\frac{2}{\pi } \frac{1}{\ell } \left( \frac{\pi }{2}+\frac{C}{\ell }\right) +\frac{2}{\pi \ell } \frac{\cos (C(2+1/\ell ))+\sin (\ell \pi )}{1+2\ell }\\&\quad +O\left( \ell ^{-2} \log \ell \right) \\&= \ell ^{-1} +O\left( \ell ^{-2} \log \ell \right) . \end{aligned}$$
\(\square \)
Proof of Lemma 4.3
We start by observing that the terms \(A_{i,\ell }\) can be written in the form
$$\begin{aligned} A_{i,\ell }=\int _{C/\ell }^{\pi /2} \frac{N_{i,\ell }(\phi )}{(1-4 \alpha ^2_{2,\ell }(\phi )/ \lambda _{\ell }^2)^{m/2}(1-4 \alpha ^2_{1,\ell }(\phi )/ \lambda _\ell ^2)^{n/2}} \sin \phi d \phi \end{aligned}$$
for a suitable function \(N_{i,\ell }(\phi )\) and \(n,m=1,2,3\). By expanding in power series around the origin the ratio
$$\begin{aligned} \frac{1}{(1-4 \alpha ^2_{2,\ell }(\phi )/ \lambda _{\ell }^2)^{m/2}(1-4 \alpha ^2_{1,\ell }(\phi )/ \lambda _\ell ^2)^{n/2}} \end{aligned}$$
and with computations analogous to those performed in the proof of Lemma 4.2, it follows that the dominant terms of \(A_{i,\ell }\) are all of the form
$$\begin{aligned} \int _{C/\ell }^{\pi /2} N_{i,\ell }(\phi ) \sin \phi d \phi . \end{aligned}$$
(10.3)
We now study the asymptotic behavior of (10.3), for \(i=1,\ldots ,8\):
-
To obtain the asymptotic behavior of \(A_{1,\ell }(\phi )\), we note that
$$\begin{aligned} \int _{C/\ell }^{\pi /2} N_{1,\ell }(\phi ) \sin \phi d \phi&=-\frac{16}{\lambda _\ell ^3} \int _{C/\ell }^{ \pi / 2} \beta _{2,\ell }^2(\phi ) \sin \phi \; d \phi . \end{aligned}$$
Now Lemma 9.5 gives the asymptotic behavior of the terms of the integrand function and Lemma 9.6 gives the asymptotic behavior of the integrand of each term, so that we get:
$$\begin{aligned} \int _{C/\ell }^{\pi /2} \Big (\frac{\beta _{2,\ell }(\phi )}{\ell ^3}\Big )^2 \sin \phi d \phi = O \Big ( \int _{C/\ell }^{\pi /2} \frac{1}{\ell ^3 \sin ^3 \phi } \sin \phi d \phi \Big )=O(\ell ^{-2}) . \end{aligned}$$
-
For the term \(A_{2,\ell }(\phi )\) we note that
$$\begin{aligned}&\int _{C/\ell }^{\pi /2} N_{2,\ell }(\phi ) \sin \phi d \phi =-\frac{16}{\lambda _\ell ^3} \int _{C/\ell }^{ \pi / 2} \beta _{1,\ell }^2 (\phi ) \sin \phi \; d \phi \end{aligned}$$
and, again applying Lemma 9.5 and Lemma 9.6, we have
$$\begin{aligned} \int _{C/\ell }^{\pi /2} \Big (\frac{ \beta _{1,\ell }(\phi )}{\ell ^3}\Big )^2 \sin \phi d \phi = O \Big ( \int _{C/\ell }^{\pi /2} \frac{1}{\ell ^3 \sin ^3 \phi } \sin \phi d \phi \Big )=O(\ell ^{-2}). \end{aligned}$$
-
The term \(A_{3,\ell }(\phi )\) leads to
$$\begin{aligned} \int _{C/\ell }^{ \pi / 2} N_{3,\ell }(\phi ) \sin \phi d \phi =-\frac{16}{\lambda _\ell ^3} \int _{C/\ell }^{ \pi / 2} \beta _{3,\ell }^2(\phi ) \sin \phi \; d \phi \end{aligned}$$
and, applying Lemma 9.5 and Lemma 9.6,
$$\begin{aligned} \int _{C/\ell }^{ \pi / 2} \Big (\frac{ \beta _{3,\ell }(\phi )}{\ell ^3}\Big )^2 \sin \phi \; d \phi = O\Big ( \int _{C/\ell }^{ \pi / 2} \frac{1}{\ell \sin \phi } \sin \phi \; d \phi \Big )=O(\ell ^{-1}). \end{aligned}$$
Since it is a dominant term, we now compute the leading constant of the term \(A_{3,\ell }(\phi )\). Recalling the definition of \(\beta _{3,\ell }\) and Lemma 9.3, we get
$$\begin{aligned}&-\frac{16}{\lambda _\ell ^3} \int _{C/\ell }^{ \pi / 2} \beta _{3,\ell }^2(\phi ) \sin \phi \; d \phi \\&\quad =-\frac{16}{\lambda _\ell ^3} \int _{C/\ell }^{ \pi / 2} \left[ -\sin ^3 \phi P'''_\ell (\cos \phi )+3 \sin \phi \cos \phi P''_\ell (\cos \phi )\right. \nonumber \\&\quad \quad \left. +\sin \phi P'_\ell (\cos \phi )\right] ^2 \sin \phi \; d \phi \\&\quad =-\frac{16}{\ell ^6} \int _{C/\ell }^{ \pi / 2} \left[ -\sin ^3 \phi \sqrt{\frac{2}{\pi }} \frac{\ell ^{3-1/2}}{\sin ^{3+1/2} \phi } \cos \psi ^+_\ell \right] ^2 \sin \phi \; d \phi +O(\ell ^{-2} \log \ell ), \end{aligned}$$
where we have also applied Lemma 9.6, Lemma 9.7 and Lemma 9.8 to identify the leading term. Now explicitly computing the integral, we have
$$\begin{aligned} -\frac{16}{\lambda _\ell ^3} \int _{C/\ell }^{ \pi / 2} \beta _{3,\ell }^2(\phi ) \sin \phi \; d \phi&=-16 {\frac{2}{\pi }}\frac{1}{\ell } \int _{C/\ell }^{ \pi / 2} \cos ^2 \psi ^+_\ell \; d \phi +O\left( \ell ^{-2} \log \ell \right) \\&=-16 {\frac{2}{\pi }} \frac{1}{\ell } \int _{C/\ell }^{ \pi / 2} \cos ^2 \left[ (\ell +1/2) \phi +\pi /4\right] \,d \phi \\&\quad +O\left( \ell ^{-2} \log \ell \right) \\&=-16 {\frac{2}{\pi }} \frac{1}{\ell } \frac{\pi }{4} +O\left( \ell ^{-2} \log \ell \right) \\&=-8 \ell ^{-1} +O\left( \ell ^{-2} \log \ell \right) . \end{aligned}$$
-
The term \(A_{4,\ell }(\phi )\) leads to
$$\begin{aligned} \int _{C/\ell }^{ \pi / 2} N_{4,\ell }(\phi ) \sin \phi d \phi =-\frac{16}{\lambda _\ell ^3} \int _{C/\ell }^{ \pi / 2} \beta _{2,\ell }(\phi ) \beta _{3,\ell }(\phi ) \sin \phi \; d \phi \end{aligned}$$
again, by Lemma 9.5 and Lemma 9.6, we have
$$\begin{aligned} \int _{C/\ell }^{\pi /2} \frac{\beta _{2,\ell }(\phi )}{\ell ^3} \frac{\beta _{3,\ell }(\phi )}{\ell ^3} \sin \phi d \phi = O \Big ( \int _{C/\ell }^{\pi /2} \frac{1}{\ell ^2 \sin ^2 \phi } \sin \phi d \phi \Big )=O(\ell ^{-2} \log \ell ). \end{aligned}$$
-
For \(A_{5,\ell }(\phi )\) we have
$$\begin{aligned} \int _{C/\ell }^{ \pi / 2} N_{5,\ell }(\phi ) \sin \phi \; d \phi&=\frac{8}{\lambda _\ell ^2} \int _{C/\ell }^{ \pi / 2} \gamma _{1,\ell }(\phi ) \sin \phi \; d \phi \\&\quad + \frac{8 \cdot 4}{\lambda _\ell ^4} \int _{C/\ell }^{ \pi / 2} \alpha _{2,\ell }(\phi ) \beta _{2,\ell }^2(\phi ) \sin \phi \; d \phi \end{aligned}$$
and then, by Lemma 9.5 and Lemma 9.7,
$$\begin{aligned}&\int _{C/\ell }^{\pi /2} \frac{\gamma _{1,\ell }(\phi )}{\ell ^{4}} \sin \phi d \phi \\&\quad = O \left( \int _{C/\ell }^{\pi /2} \frac{1}{\ell ^{2+1/2} \sin ^{2+1/2} \phi } \sin \phi d \phi \right) =O(\ell ^{-2}),\\&\quad \int _{C/\ell }^{\pi /2} \Big ( \frac{\beta _{2,\ell }(\phi )}{\ell ^3}\Big )^{2} \frac{\alpha _{2,\ell }(\phi )}{\ell ^2} \sin \phi d \phi \\&\quad = O \Big ( \int _{C/\ell }^{\pi /2} \frac{1}{\ell ^{3+1/2} \sin ^{3+1/2} \phi } \sin \phi d \phi \Big )=O(\ell ^{-2}). \end{aligned}$$
-
The term \(A_{6,\ell }(\phi )\) leads to
$$\begin{aligned} \int _{C/\ell }^{ \pi / 2} N_{6,\ell }(\phi ) \sin \phi \; d \phi&=\frac{8}{\lambda _\ell ^2} \int _{C/\ell }^{ \pi / 2} \gamma _{2,\ell }(\phi ) \sin \phi \; d \phi \\&\quad + \frac{8 \cdot 4}{\lambda _\ell ^4} \int _{C/\ell }^{ \pi / 2} \alpha _{1,\ell }(\phi ) \beta _{1,\ell }^2(\phi ) \sin \phi \; d \phi \end{aligned}$$
and again applying Lemma 9.5 and Lemma 9.7, we have
$$\begin{aligned}&\int _{C/\ell }^{\pi /2} \frac{\gamma _{2,\ell }(\phi )}{\ell ^{4}} \sin \phi d \phi = O \Big ( \int _{C/\ell }^{\pi /2} \frac{1}{\ell ^{1+1/2} \sin ^{1+1/2} \phi } \sin \phi d \phi \Big )=O(\ell ^{-1-1/2}),\\&\int _{C/\ell }^{\pi /2} \Big (\frac{\beta _{1,\ell }(\phi )}{\ell ^3}\Big )^2 \frac{\alpha _{1,\ell }(\phi )}{\ell ^2} \sin \phi d \phi = O \Big ( \int _{C/\ell }^{\pi /2} \frac{1}{\ell ^{4+1/2} \sin ^{4+1/2} \phi } \sin \phi d \phi \Big )\\&\quad =O(\ell ^{-2}). \end{aligned}$$
-
For \(A_{7,\ell }(\phi )\), we get
$$\begin{aligned} \int _{C/\ell }^{ \pi / 2} N_{7,\ell }(\phi ) \sin \phi \; d \phi&=\frac{8}{\lambda _\ell ^2} \int _{C/\ell }^{ \pi / 2} \gamma _{4,\ell }(\phi ) \sin \phi \; d \phi \\&\quad + \frac{8 \cdot 4}{\lambda _\ell ^4} \int _{C/\ell }^{ \pi / 2} \alpha _{2,\ell }(\phi ) \beta _{3,\ell }^2(\phi ) \sin \phi \; d \phi \end{aligned}$$
and, by Lemma 9.5 and Lemma 9.7,
$$\begin{aligned}&\int _{C/\ell }^{\pi /2} \frac{ \gamma _{4,\ell }(\phi )}{\ell ^{4}} \sin \phi d \phi \nonumber \\&\quad = O \Big ( \int _{C/\ell }^{\pi /2} \frac{1}{\ell ^{1/2} \sin ^{1/2} \phi } \sin \phi d \phi \Big )=O(\ell ^{-1/2}),\nonumber \\&\quad \int _{C/\ell }^{\pi /2} \Big (\frac{ \beta _{3,\ell }(\phi )}{\ell ^3}\Big )^2 \frac{\alpha _{2,\ell }(\phi )}{\ell ^2} \sin \phi d \phi \nonumber \\&\quad = O \Big ( \int _{C/\ell }^{\pi /2} \frac{1}{\ell ^{1+1/2} \sin ^{1+1/2} \phi } \sin \phi d \phi \Big )=O\left( \ell ^{-1-1/2}\right) . \end{aligned}$$
(10.4)
Since the leading term in \(\gamma _{4,\ell }(\phi )\) is oscillatory, we can get a sharper bound for the term in (10.4), by observing that
$$\begin{aligned}&\int _{C/\ell }^{\pi /2} \frac{ \gamma _{4,\ell }(\phi )}{\ell ^{4}} \sin \phi d \phi \\&\quad = \frac{ 1}{\ell ^{4}} \int _{C/\ell }^{\pi /2} \left[ \sin ^{4}\phi P_{\ell }^{\prime \prime \prime \prime }(\cos \phi )-6\sin ^{2}\phi \cos \phi P_{\ell }^{\prime \prime \prime }(\cos \phi )\right. \\&\quad \quad \left. +\left( -4\sin ^{2}\phi +3\cos ^{2}\phi \right) P_{\ell }^{\prime \prime }(\cos \phi )+\cos \phi P_{\ell }^{\prime }(\cos \phi )\right] \sin \phi d \phi \\&\quad = \frac{1}{\sqrt{\ell }} \int _{C/\ell }^{\pi /2} \frac{1}{ \sin ^{1/2} \phi } \left[ \cos \psi ^-_\ell - \frac{1}{8 \ell \phi } \cos \psi ^+_\ell \right] \sin \phi d \phi +O\left( \ell ^{-1-1/2}\right) \\&\quad \le \frac{1}{\sqrt{\ell }} \int _{C/\ell }^{\pi /2} \cos \psi ^-_\ell d \phi + \frac{1}{\sqrt{\ell }} \int _{C/\ell }^{\pi /2} \frac{1}{8 \ell \phi } d \phi +O(\ell ^{-1-1/2}) \\&\quad = \frac{1}{\sqrt{\ell }} \frac{2}{1+2 \ell } \left[ \sin \left( \frac{\ell \pi }{2}\right) + \sin \left( \frac{\pi }{4} - \frac{C}{2 \ell } -C\right) \right] + O\left( \ell ^{-1-1/2}\right) \\&\quad =O\left( \ell ^{-1-1/2}\right) . \end{aligned}$$
-
Finally, for \(A_{8,\ell }(\phi )\) we have
$$\begin{aligned} \int _{C/\ell }^{ \pi / 2} N_{8,\ell }(\phi ) \sin \phi \; d \phi&=\frac{8}{\lambda _\ell ^2} \int _{C/\ell }^{ \pi / 2} \gamma _{3,\ell }(\phi ) \sin \phi \; d \phi \\&\quad + \frac{8 \cdot 4}{\lambda _\ell ^4} \int _{C/\ell }^{ \pi / 2} \alpha _{2,\ell }(\phi ) \beta _{2,\ell }(\phi ) \beta _{3,\ell }(\phi ) \sin \phi \; d \phi \end{aligned}$$
where, from Lemma 9.5 and Lemma 9.7, we have
$$\begin{aligned} \int _{C/\ell }^{\pi /2} \frac{ \gamma _{3,\ell }(\phi )}{\ell ^{4}} \sin \phi d \phi&= O \Big ( \int _{C/\ell }^{\pi /2} \frac{1}{\ell ^{1+1/2} \sin ^{1+1/2} \phi } \sin \phi d \phi \Big )\\&=O(\ell ^{-1-1/2}),\\ \int _{C/\ell }^{\pi /2} \frac{\beta _{2,\ell }(\phi )}{\ell ^{3}} \frac{\beta _{3,\ell }(\phi ) }{\ell ^{3}} \frac{\alpha _{2,\ell }(\phi )}{\ell ^2} \sin \phi d \phi&= O \Big ( \int _{C/\ell }^{\pi /2} \frac{1}{\ell ^{2+1/2} \sin ^{2+1/2} \phi } \sin \phi d \phi \Big )\\&=O(\ell ^{-2}). \end{aligned}$$
\(\square \)
Proof of Lemma 4.4
We note that the second-order terms are all of the form
$$\begin{aligned} \int _{C/\ell }^{ \pi / 2} \frac{a_{i,\ell }(\phi ) a_{j,\ell }(\phi ) }{ \sqrt{ (1 -4 \alpha _{2,\ell }^2/\lambda _\ell ^2) (1 -4 \alpha _{1,\ell }^2/\lambda _\ell ^2) }}\sin \phi \; d \phi \end{aligned}$$
with \(i,j=1,2, \ldots ,8\). Applying Lemma 9.6, Lemma 9.7 and Lemma 9.8 we identify, as in the proof of the previous lemma, that the product of two terms \(a_{i,\ell }(\phi ) a_{j,\ell }(\phi )\) such that at least one is non-dominant produces a non-dominant term, so that, if \((i,j) \ne (7,7)\), we immediately see that
$$\begin{aligned} \int _{C/\ell }^{ \pi / 2} \frac{a_{i,\ell }(\phi ) a_{j,\ell }(\phi ) }{ \sqrt{ (1 -4 \alpha _{2,\ell }^2/\lambda _\ell ^2) (1 -4 \alpha _{1,\ell }^2/\lambda _\ell ^2) }}\sin \phi \; d \phi&= O \Big (\int _{C/\ell }^{ \pi / 2} \frac{1}{\ell ^2 \sin ^2 \phi } \sin \phi \; d \phi \Big )\\&=O(\ell ^{-2} \log \ell ). \end{aligned}$$
Instead, for \((i,j) = (7,7)\), we have the square of the integrand function in (10.4), that in view of Lemma 9.6, is immediately seen to be dominant, i.e.,
$$\begin{aligned}&\int _{C/\ell }^{ \pi / 2} \frac{a^2_{7,\ell }(\phi ) }{ \sqrt{ (1 -4 \alpha _{2,\ell }^2/\lambda _\ell ^2) (1 -4 \alpha _{1,\ell }^2/\lambda _\ell ^2) }} \sin \phi \; d \phi \\&\quad =O\Big (\int _{C/\ell }^{ \pi / 2} \frac{1}{\ell \sin \phi } \sin \phi \; d \phi \Big )=O( \ell ^{-1}). \end{aligned}$$
Since we need the multiplicative constant of the leading terms we first note that:
$$\begin{aligned} a^2_{7,\ell }(\phi )&= \frac{8^2}{\lambda _\ell ^4} \left[ \gamma _{4,\ell }^2(\phi )+\frac{16}{(1 -4 \alpha _{2,\ell }^2(\phi )/\lambda _\ell ^2)^2} \frac{\alpha _{2,\ell }^2(\phi ) \beta _{3,\ell }^4(\phi )}{\lambda _\ell ^4}\right. \\&\quad \left. - \frac{8}{1 -4 \alpha _{2,\ell }^2(\phi )/\lambda _\ell ^2} \frac{\gamma _{4,\ell }(\phi ) \alpha _{2,\ell }(\phi ) \beta _{3,\ell }^2(\phi )}{\lambda _\ell ^2} \right] , \end{aligned}$$
then, by isolating the leading integral terms with the aid of Lemma 9.6, Lemma 9.7 and Lemma 9.8, and finally by computing the integral, we get
$$\begin{aligned}&\int _{C/\ell }^{ \pi / 2} a^2_{7,\ell }(\phi ) \sin \phi \; d \phi \\&\quad =8^2\int _{C/\ell }^{ \pi / 2} \frac{\gamma _{4,\ell }^2(\phi )}{\lambda _\ell ^4} \sin \phi d \phi +O \Big ( \int _{C/\ell }^{ \pi / 2} \frac{1}{\ell ^2 \sin ^2 \phi } \sin \phi d \phi \Big ) \\&\quad =8^2 \int _{C/\ell }^{ \pi / 2} \left\{ \sqrt{\frac{\pi }{2}} \frac{1}{\ell ^{1/2} \sin ^{1/2} \phi } \left[ \cos \psi ^-_\ell -\frac{1}{8 \ell \phi } \cos \psi ^+_\ell \right] \right. \\&\quad \quad \left. +O\left( \frac{1}{\ell ^{1+1/2} \sin ^{1+1/2} \phi }\right) \right\} ^2 \sin \phi d \phi + O \left( \int _{C/\ell }^{ \pi / 2} \frac{1}{\ell ^2 \sin ^2 \phi } \sin \phi d \phi \right) \\&\quad =8^2 \int _{C/\ell }^{ \pi / 2} \left\{ \sqrt{\frac{\pi }{2}} \frac{1}{\ell ^{1/2} \sin ^{1/2} \phi } \left[ \cos \psi ^-_\ell -\frac{1}{8 \ell \phi } \cos \psi ^+_\ell \right] \right\} ^2 \sin \phi d \phi \\&\quad \quad + O \Big ( \int _{C/\ell }^{ \pi / 2} \frac{1}{\ell ^2 \sin ^2 \phi } \sin \phi d \phi \Big )\\&\quad =8^2 \int _{C/\ell }^{ \pi / 2} \left\{ \frac{2}{\pi } \frac{1}{\ell \sin \phi } \left[ \frac{1}{2} + \frac{1}{2} \cos (2 \psi ^-_\ell )+ \frac{1}{64 \ell ^2 \phi ^2} \cos ^2 \psi ^+_\ell \right. \right. \\&\quad \quad \left. \left. - \frac{1}{4 \ell \phi } \cos \psi ^-_\ell \cos \psi ^+_\ell \right] \right\} \sin \phi d \phi + O \Big ( \int _{C/\ell }^{ \pi / 2} \frac{1}{\ell ^2 \sin ^2 \phi } \sin \phi d \phi \Big )\\&\quad = {32} \ell ^{-1}+O(\ell ^{-2} \log \ell ). \end{aligned}$$
\(\square \)
Appendix 5: Nonsingularity of the Covariance Matrix for \(\phi <c/\ell \)
We only need to show that, after scaling, the determinant of the matrix \(A_\ell (\psi )\), evaluated for points on the equatorial line \(x=(\pi /2,\phi ),y=(\pi /2,0)\), is strictly positive for \(c>\psi >0\); for points outside the equator the covariance matrix is obtained by a change of basis: the corresponding matrix does not depend on \(\ell \) and can be easily shown to be full rank.
By expanding the terms of the matrix up to order 4 around \(\psi =0\), for \(\lambda _\ell =\ell (\ell +1)\) we have
$$\begin{aligned} \alpha _{1,\ell }(\psi )&=\frac{\ell +1}{2\ell }-\frac{ \lambda _{\ell } ( \lambda _{\ell }-2 ) \psi ^{2}}{16\, \ell ^{4}}\\&\quad +\frac{ (\ell -1)(\ell +1)(\ell +2) ( \lambda _{\ell }-4 ) \psi ^{4}}{2^7 \, 3\, \ell ^{5}} +O ( \psi ^{6} )\\ \alpha _{2,\ell }(\psi )&=\frac{\ell +1}{2 \ell }-\frac{\lambda _{\ell } (3 \, \lambda _{\ell }-2) \psi ^2}{2^4 \, \ell ^4}\\&\quad +\frac{\lambda _{\ell } (5 \, (\ell -1) \ell (\ell +1) (\ell +2) +2^3) \psi ^4}{2^7 \, 3\, \ell ^6} +O (\psi ^{6} ). \end{aligned}$$
It should be noted that, as \(\ell \rightarrow \infty \), all coefficients converge to constants; more importantly, the constants involved in the O-notation for all the \(O (\psi ^{6}) \) terms are universal. A computer-oriented computation yields the following Taylor expansion for the determinant:
$$\begin{aligned} \text {det}(A_{\ell }(\psi ))&=\frac{ (\ell -1)(\ell +1)^{4}(\ell +2) (3 \ell (\ell +1) -2) \psi ^{4}}{2^{8}\; \ell ^{8}}+O( \psi ^{6})\\&=\psi ^{4} \Big (\frac{3}{2^8}+O(\ell ^{-1}) \Big )+O(\psi ^{6})>0 \end{aligned}$$
the inequality holding for c sufficiently small, because by the above, the \(O(\psi ^{6})\) term is universal.
