Haar and Shannon Wavelets on Certain Locally Compact Abelian Groups

Haar and Shannon wavelets are the simple and fundamental wavelets on L 2 (R).On a Euclidean domain, both the wavelets are at poles apart in the way because, Haar wavelet basis is well localized in space but not in frequency in contrast to Shannon wavelet basis which is well localized in frequency domain but not in time domain.In this paper, Haar and Shannon wavelets are constructed on a locally compact Abelian group G having open compact subgroup, using multiresolution analysis and established that both are same on G. Also, the proposed construction is illustrated using examples of locally compact Abelian groups.