The Continuous Wavelet Transform in Sobolev Spaces Over Locally Compact Abelian Group and Its Approximation Properties

Abstract

The Sobolev space over locally compact abelian group Hs(G) is defined and we extend the continuous wavelet transform to Sobolev space Hs(G) for arbitrary real s. This generalisation of the wavelet transform naturally leads to a unitary operator between these spaces. Further, the asymptotic behaviour of the transforms of the L2 function for small scaling parameters is examined. In special cases, the wavelet transform converges to a generalized derivative of its argument. We also discuss the consequences for the discrete wavelet transform arising from this property. Numerical examples illustrate the main result.