In this paper the theory of wavelets on the integers is developed. For this, one needs to first find analogs of translations and dyadic dilations which appear in the classical theory. Translations in l2(Z) are defined in the obvious way, taking advantage of the additive group structure of the integers. Dyadic dilations, on the other hand, pose a greater problem. In the classical theory of wavelets on the real line, translation T and dyadic dilation T obey the commutativity relation DT^2 = TD. We choose to define dyadic dilations on the integers in terms of this functional equation. All such dyadic dilations are characterized and the corresponding multiresolution structures they generate are introduced and examined. The main results of this paper focus on connecting multiresolution structures and wavelets on the integers with their counterparts on the line and include the fact that every wavelet on the integers is a MRA wavelet.
@article{urn:eudml:doc:42782,
title = {Wavelets on the integers.},
journal = {Collectanea Mathematica},
volume = {52},
year = {2001},
pages = {257-288},
zbl = {0996.42024},
mrnumber = {MR1885222},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:42782}
}
Gressman, Philip. Wavelets on the integers.. Collectanea Mathematica, Tome 52 (2001) pp. 257-288. http://gdmltest.u-ga.fr/item/urn:eudml:doc:42782/