= Solution
Write $S_2(r)=\langle(\Delta v)^2\rangle$ and normalize the <turbulent energy spectrum> by $\int_0^\infty E(k)\,dk=\langle|\mathbf u|^2\rangle/2$. In isotropic incompressible <turbulence> the <longitudinal velocity correlation> is
$$
Q_{LL}(r)=2\int_0^\infty E(k)\frac{\sin(kr)-kr\cos(kr)}{(kr)^3}\,dk.
$$
Combining $S_2=2[Q_{LL}(0)-Q_{LL}(r)]$ with $Q_{LL}(0)=2\int E\,dk/3$ gives the exact spectral filter
$$
\boxed{\frac34S_2(r)=\int_0^\infty E(k)H(kr)\,dk},\qquad
H(x)=1+\frac{3\cos x}{x^2}-\frac{3\sin x}{x^3}.
$$
The expression has a smooth removable limit at $x=0$. Its asymptotic forms are
$$
H(x)=\frac{x^2}{10}-\frac{x^4}{280}+O(x^6)\quad(x\ll1),\qquad
H(x)=1+O(x^{-2})\quad(x\gg1).
$$
An eddy much larger than $r$ contributes nearly the same velocity at the two points and mostly cancels in the increment. An eddy much smaller than $r$ contributes decorrelated velocities and does not cancel. Replacing the smooth transition of $H$ by a step at $kr=\pi$ therefore gives the common estimate $3S_2/4\simeq\int_{\pi/r}^\infty E(k)\,dk$. The cutoff corresponds roughly to a half wavelength equal to the point separation; its exact numerical location is conventional.
Keeping the leading nonzero low-wavenumber term gives the improved estimate
$$
\boxed{\frac34S_2(r)\simeq\int_{\pi/r}^\infty E(k)\,dk+
\frac{r^2}{10}\int_0^{\pi/r}k^2E(k)\,dk}.
$$
It remains imperfect because $kr$ is not small throughout the whole low-wavenumber interval and the exact kernel oscillates above the transition. The data sheet's alternative coefficient $1/\pi^2$ is a nearby approximate matching prescription; $1/10$ is the exact Taylor coefficient. Neither piecewise filter is an exact identity.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-73-structure-function-filter.png]
{title=Exact longitudinal structure-function spectral filter and two cutoff approximations}
Higher-order <longitudinal structure functions> also retain large-scale information. Decompose $\mathbf u=\mathbf U+\mathbf u_s$, where $\mathbf U$ varies on a length much larger than $r$. Its longitudinal increment is $\Delta U_L\simeq r\,\partial_LU_L$, which is small but not zero. Expanding $(\Delta U_L+\Delta u_{s,L})^p$ gives its own <moments> and mixed <moments> with the small-scale increment. Rare large gradients or correlated amplitude modulation can matter particularly strongly at large $p$.
Thus \b[a velocity increment is not an exact scale-local bandpass filter]. Its <moments> may retain large-scale strain and forcing information, undermining an unqualified claim of universal small-scale statistics determined by one local parameter. This observation alone does not refute an asymptotic <inertial range> theory: for a three-dimensional $k^{-5/3}$ spectrum, the low-wavenumber correction is mainly supplied by wavenumbers of order $1/r$, while an isolated integral-scale smooth contribution behaves as $r^2$ and is subleading to $r^{2/3}$. The issue is whether large-scale contamination and mixed <moments> actually become negligible in the regime and <moment> order being used.
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