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Apply the isotropic cosmological correlation-power-spectrum relation and put $K=k_{\max }$, $q=Kr$. The band-limited linear density correlation isTwo integrations by parts give the primitive $-k^2\cos (kr)/r+2k\sin (kr)/r^2+2\cos (kr)/r^3$. Evaluating both endpoints yieldsThe apparent singularity at the origin is removable. Either the Taylor series or the original integral givesIn fact $\xi (r)=\xi (0)[1-q^2/9+q^4/240+O(q^6)]$. The sharp spectral cutoff produces oscillations and negative correlations at some nonzero separations, even though the matter power spectrum is everywhere nonnegative. A covariance function need not be pointwise nonnegative; its required positivity is that of the covariance quadratic form.

Write $K=k_{\max }r$. Using the transform in part (a), the band-limited linear density correlation isTwo integrations by parts give the antiderivative $-k^2\cos (kr)/r+2k\sin (kr)/r^2+2\cos (kr)/r^3$. Subtract its value at zero to obtain, for $r>0$,The origin is a removable limit. The bracket starts with $K^4/4$, soThis is also the density variance obtained directly by setting $\sin (kr)/(kr)=1$ in the integral. The sharp spectral edge produces oscillatory correlations, with leading large-$r$ term $-A{k}_{\max }^2\cos (k_{\max }r)/(2{\pi }^2{r}^2)$. A negative correlation at some separation is compatible with a nonnegative power spectrum.