# Wavelet transform of a spatial convolution

Does anyone know if there exist a kind of convolution theorem for the discrete wavelet transform (decimated or undecimated)?

In other words can I find a simple form of $$W\left[ \int f(t) g(x-t) \, dt\right]$$ where $$W$$ is the discrete wavelet transform operator?

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 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.