The wavelet transform on Sobolev spaces and its approximation properties

Summary

We extend the continuous wavelet transform to Sobolev spacesH ^s(ℜ) for arbitrary reals and show that the transformed distribution lies in the fiber spaces$L_2 \left( {\left( {\mathbb{R}_0 ,\frac{{da}}{{a^2 }}} \right),H^s \left( \mathbb{R} \right)} \right) \cong H^{0,s} \left( {\mathbb{R}^2 ,\frac{{dadb}}{{a^2 }}} \right)$. This generalisation of the wavelet transform naturally leads to a unitary operator between these spaces.

Further the asymptotic behaviour of the transforms ofL ₂-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.