  We analyse the Gaussian wave packet transform. Based on the Fourier inversion
formula and a partition of unity, which is formed by a collection of Gaussian
basis functions, a new representation of square-integrable functions is
presented. Including a rigorous error analysis, the variants of the wave packet
transform are then derived by a discretisation of the Fourier integral via
different quadrature rules. Based on Gauss--Hermite quadrature, we introduce a
new representation of Gaussian wave packets in which the number of basis
functions is significantly reduced. Numerical experiments in 1D illustrate the
theoretical results.
