Towards a General Description of Translation-Invariant and Translation-Covariant Linear Transformations: A Natural Justification of Fourier Transforms and Fuzzy Transforms
Irina Perfilieva, Владик Крейнович · scholarworks - UTEP (The University of Texas at El Paso) · 2011
In many practical situations, we are interested in the dependencies that do not change with time, i.e., that do not change when we change the origin of the time axis. The corresponding translation-invariant transformations are easy to describe: they correspond to convolutions, or, equivalently, to fuzzy transforms. It turns out that if we relax the invariance condition and require only that the transformation be translation-covariant (i.e., that it appropriately changes under translation), we get exactly two classes of transformations: Fourier transforms and fuzzy transforms. This result explain why both transforms have been successfully used in data processing.