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 。