The oscillatory cancellation mechanism of translation orthogonality, with an application to gaussian-modulated shannon wavelets

The integer-translation orthogonality of the Shannon kernel sinc is not a consequence of spatial separation of the two factors: the supports of sinc(t) and sinc(t – 1) overlap on a set of positive measure. This paper identifies the oscillatory cancellation mechanism as the structural source of translation orthogonality, and shows that it is broken by Gaussian modulation. For the family introduced by Wei and collaborators in the late 1990s as a time-localized alternative to the Shannon wavelet, we prove that Gaussian modulation, although smooth and arbitrarily weak, reweights the positive and negative lobes asymmetrically and destroys the cancellation, and the proof rests on a direct time-domain analysis of the oscillatory sign structure of the symmetrized integrand. To the best of our knowledge, no rigorous proof of either the preservation or the breakdown of integer-translation orthogonality has been given in the nearly three decades since these wavelets were introduced, despite the elementary formulation of the question and the central role of these kernels in numerical analysis (most prominently for the viscous Burgers equation) and in computational chemistry.