A wavelet theory for local fields and related groups

Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G = Q(p), the field of p-adic rational numbers (as a group under addition), which has compact open subgroup H = Z(p), the ring of p-adic integers. Classical wavelet theories, which require a non trivial discrete subgroup for translations, do riot apply to G, which may not have such a subgroup. A wavelet theory is developed art G using coset representatives of the discrete quotient (G) over cap /H-perpendicular to to circumvent this limitation. Wavelet bases are constructed by means of an iterative method giving rise to so-called wavelet sets in the dual group (G) over cap. Although the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, it is observed that their analogues for G are equivalent.