  It is shown that any convolution operator in the time domain can be
represented exactly as a multiplication operator in the time-scale (wavelet)
domain. The Mellin transform gives a one-to-one correspondence between
frequency filters (multiplications in the frequency domain) and scale filters
(multiplications in the scale domain), subject to the convergence of the
defining integrals. The usual wavelet reconstruction theorem is a special case.
Applications to the denoising of random signals are proposed. It is argued that
the present method is more suitable for removing the effects of atmospheric
turbulence than the conventional procedures because it is ideally suited for
resolving spectral power laws.
