We analyze 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 discretization 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.

翻译：我们分析高斯波包的变换。 根据由高斯基函数集合而成的Fourier 反向公式和统一分割法, 展示了可辨别函数的新表示。 包括严格的错误分析, 包括波包变换的变异通过不同二次曲线规则的Fourier元集分离产生。 根据高斯- Hermite 二次曲线, 我们引入了高斯波包的新表示法, 基函数的数量会大大减少。 1D 中的数值实验显示了理论结果 。