The wavelet transform on Sobolev spaces and its approximation properties

Summary We extend the continuous wavelet transform to Sobolev spacesH^s() for arbitrary reals and show that the transformed distribution lies in the fiber spaces. This generalisation of the wavelet transform naturally leads to a unitary operator between these spaces.Further the asymptotic behaviour of the transforms ofL₂-functions for small scaling parameters is examined. In special cases the wevelet 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.Supported by the Deutsche Forschungsgemeinschaft under grant Lo 310/2-4