Unaliased fixed-grid spectral bias under the exact no-alias condition

By [F1] the sampled field has torus density $g_{\delta }$, and the assumed representative meets all hypotheses of [F2]. Thus the denominator in [F3] is eventually positive and both angular expectations tend to $g_{\delta }({\lambda }_0)$. Multiplying by ${\delta }^d$ and reindexing the convergent nonnegative sum at the target gives ${\delta }^d{g}_{\delta }({\lambda }_0)={\sum }_{k}f({\omega }_0+2\pi k\mathrm{/}\delta )=a_{\delta }({\omega }_0)$. This proves the general aliased limit for both physically normalized estimates.

Every alias summand is nonnegative. Therefore $a_{\delta }({\omega }_0)-f({\omega }_0)={\sum }_{k\mathrm{\ne }0}f({\omega }_0+2\pi k\mathrm{/}\delta )$ is zero if and only if every summand is zero. This proves both directions of the stated exact condition. Under the sufficient global condition, only the central translate is nonzero for frequencies in the Nyquist cube; its support stays away from the boundary. Hence the periodic density is bounded and continuous across the boundary as well, and has no noncentral alias at the stated target. All hypotheses and the exact condition are then verified.