Wavelet transform of a spatial convolution
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?


The nonexistence of a wavelet function admitting a wavelet convoluton theorem of the Fourier type, 1994, A. R. Lindsey, unpublished report
Convolution using the undecimated discrete wavelet transform, 1996



  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.



Convolution theorems for linear transforms, 1998



  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.



The convolution theorem for the continuous wavelet tranform, 2004



  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.



On the uniqueness of the convolution theorem for the Fourier transform, 2008



  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.