论文标题
Hermite功能和傅立叶系列
Hermite functions and Fourier series
论文作者
论文摘要
使用归一化的HERMITE函数，我们在单位圆（$ l^2（\ Mathcal C）$）和$ L_2（\ Mathbb Z）$中构建基础，它们通过傅立叶变换和离散的傅立叶变换相互关联。这些关系是统一的。正统碱基的构建需要使用Gramm-schmidt方法。在这两个空间上，我们为梯子运算符提供了与一维量子振荡器相同的属性。这些运算符是某些类似乘法和分化的操作员的线性组合，当应用于周期功能时，它们会保留周期性。最后，我们已经为$ l^2（\ Mathcal C）$和$ L_2（\ Mathbb Z）$构建了索具，因此所有提到的运营商都是连续的。
Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle ($L^2(\mathcal C)$) and in $l_2(\mathbb Z)$, which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm--Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both $L^2(\mathcal C)$ and $l_2(\mathbb Z)$, so that all the mentioned operators are continuous.
