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With the specified Fourier transform convention, the inverse is $\delta (\mathbf{x})=\int d^{3}k{\delta }_{\mathbf{k}}{e}^{-i\mathbf{k}\cdot \mathbf{x}}/(2\pi {)}^3$. Inserting both inverse transforms into the cosmological two-point correlation function and applying the given Dirac delta covariance yieldsFor an isotropic matter power spectrum, align the polar axis with $\mathbf{r}$. The angular integral is $2\pi {\int }_{-1}^1{e}^{ikr\mu }d\mu =4\pi \sin (kr)/(kr)$. Hence the isotropic cosmological correlation-power-spectrum relation isThe angular kernel is the order-zero Spherical Bessel function $j_0(kr)$. At $r=0$, replace it by its limit one, giving the unsmoothed variance when the integral converges. The argument assumes an ordinary integrable spectrum, or an explicitly specified distributional Fourier transform interpretation otherwise.

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.