THE WAVELET TRANSFORM ON SOBOLEV SPACES AND ITS APPROXIMATION PROPERTIES

We extend the continuous wavelet transform to Sobolev spaces H(s)(IR) for arbitrary real s and show that the transformed distribution lies in the fiber spaces [GRAPHICS] This generalisation of the wavelet transform naturally leads to a unitary operator between these spaces. Further the asymptotic behaviour of the transforms of L2-functions 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.