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 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.

中文翻译：

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通过求积规则进行高斯波包变换

我们分析高斯波包变换。基于傅里叶反演公式和由高斯基函数集合构成的单位分部，提出了平方可积函数的新表示。包括严格的误差分析，然后通过不同的求积规则对傅里叶积分进行离散化，得出波包变换的变体。基于高斯-埃尔米特求积，我们引入了一种新的高斯波包表示，其中基函数的数量显着减少。一维数值实验说明了理论结果。