  The connection between derivative operators and wavelets is well known. Here
we generalize the concept by constructing multiresolution approximations and
wavelet basis functions that act like Fourier multiplier operators. This
construction follows from a stochastic model: signals are tempered
distributions such that the application of a whitening (differential) operator
results in a realization of a sparse white noise. Using wavelets constructed
from these operators, the sparsity of the white noise can be inherited by the
wavelet coefficients. In this paper, we specify such wavelets in full
generality and determine their properties in terms of the underlying operator.
