I cannot say I have a clear understanding of this at this time. However, a few pointers. I'd love to see somebody provide a detailed account. Others bits at: Multiplication in the wavelet domain, what does it look like in real space?
Convolution is one of the most widely used digital signal processing
operations. It can be implemented using the fast Fourier transform
(FFT), with a computationalcomplexity of $O(N \log N)$. The
undecimated discrete wavelet transform (UDWT) is linear and shift
invariant, so it can also be used to implement convolution. In this
paper, we propose a scheme to implement the convolution using the
UDWT, and study its advantages and limitations.
This correspondence explores the existence of convolution theorem for
linear transformations under a variety of different assumptions. There
are eight convolution theorems, all Fourier-related with only N
operations in the transform domain and no ordering constraints on the
convolution components in the result. They include circular
convolutions and correlations.
We study the application of the continuous wavelet transform to
perform signal filtering processes. We first show that the convolution
and correlation of two wavelet functions satisfy the required
admissibility and regularity conditions. By using these new wavelet
functions to analyze both convolutions and correlations, respectively,
we derive convolution and correlation theorems for the continuous
wavelet transform and show them to be similar to that of other joint
spatial/spatial–frequency or time/frequency representations. We then
investigate the effect of multiplying the continuous wavelet transform
of a given signal by a related transfer function and show how to
perform spatially variant filtering operations in the wavelet domain.
Finally, we present numerical examples showing the usefulness of
applying the convolution theorem for the continuous wavelet transform
to perform signal restoration in the presence of additive noise.
This paper shows that members of the fourier transform family are the
only linear transforms that have a convolution theorem, that is, that
can replace $O(N^2)$ operations of a convolution in a time domain by
$O(N)$ operations in a transform domain. Generally, there is an
additional cost to compute the transform itself. Our observation is
motivated by recent activity in wavelet and subband decompositions and
related spectral analyses, which are attractive alternatives for
signal compression applications. A natural question when using such
techniques is to determine if convolutions of $N$-point signals can be
calculated with fewer operations in a compressed transform domain than
in an uncompressed time domain. The answer is negative for a broad set
of assumptions. This paper indicates what assumptions must be relaxed
in seeking a linear transform that has a convolution theorem
comparable to the convolution theorem for Fourier transforms.