A wavelet theory for local fields and related groups

By John J. Benedetto, Robert L. Benedetto. 2003

198 citations recorded by the source corpus.

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 padic 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 socalled 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/30a74a9ab94cd1aa9b08ac8b2898a8dee19920ce

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Computer Vision and Pattern Recognition

Citations in the corpus

On p-adic mathematical physics (2006, methodology): “Important contributions in p-adic wavelet theory were done by Albeverio, Benedetto, Khrennikov, Shelkovich, Skopina [171, 172, 173, 174, 175, 176]. p-Adic Tauberian and Shannon-Kotelnikov theorems were investigated by Khrennikov and Shelkovich [177].”
p-Adic Multiresolution Analysis and Wavelet Frames (2008, background): “Benedetto [2, 3], however, discussed if it is possible to construct other p-adic wavelets with the same set of translations.”
Wavelets on ultrametric spaces (2005, methodology): “In [5,6] by J.”
Orthogonal wavelets on direct products of cyclic groups (2007, background): “5] and a detailed bibliography is given in [2].”
p-Adic mathematical physics: the first 30 years (2017, methodology): “Important contributions in p-adic wavelet theory were done also by Albeverio, Benedetto, Khrennikov, Shelkovich, Skopina, Evdokimov [41, 42, 6, 7, 143, 144, 222, 145, 15].”
Examples of wavelets for local fields (2003, methodology): “Given our dilation operator δA and our group of translation operators G/H = {τ[s]}, we may now define wavelets on G as in [ 2 ].”