  We introduce a new type of quadrature, known as approximate Gaussian
quadrature (AGQ) rules using {\epsilon}-quasiorthogonality, for the
approximation of integrals of the form \int f(x)d \alpha(x). The measure
{\alpha}(\cdot) can be arbitrary as long as it possesses finite moments {\mu}n
for sufficiently large n. The weights and nodes associated with the quadrature
can be computed in low complexity and their count is inferior to that required
by classical quadratures at fixed accuracy on some families of integrands.
Furthermore, we show how AGQ can be used to discretize the Fourier transform
with few points in order to obtain short exponential representations of
functions.
