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 = Qp, the field of p-adic rational numbers (as a group under addition), which has compact open subgroup H = Zp, the ring of padic integers. Classical wavelet theories, which require a non-trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. A wavelet theory is developed on G using coset representatives of the discrete quotient Ĝ/H 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 Ĝ. Although the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, it is observed that their analogues for G are equivalent.