On the Non Existence of a Wavelet Function Admitting a Convolution Theorem of the Fourier Type

摘要

The Fourier and Laplace transforms, and the Z-transform in the discrete case, all boast very similar convolution theorems, relating a translation-based convolution in time, space, or position to a transform-domain product. The key to this elegant structure is the kernel of the transform integral. In the above cases this kernel is a complex exponential either on the unit circle or in the general complex plane, and possesses separability properties conducive to an elegant convolution theorem. With the advent and maturation of wavelet theory comes the natural desire for a similar result with the wavelet transform. Since there are endless varieties of wavelet functions acting as kernels for the wavelet transform, one might fathom the existence of at least one that would provide the appropriate separability properties for a simple convolution theorem. But it is shown here that no wavelet function exists which allows a translation-based convolution operation to transform into a component-wise product in the wavelet- domain.