Hermite函数和Fourier级数
Hermite functions and Fourier series
摘要：
利用正规化Hermite函数，构造了单位圆上平方可积函数空间（$L^2(\mathcal C)$）和$l_2(\mathbb Z)$中的基，并通过Fourier变换和离散Fourier变换将它们联系起来.这些关系是统一的。标准正交基的构造需要使用Gramm-Schmidt方法。在这两个空间上，我们都提供了与一维量子振子的梯形算符具有相同性质的梯形算符。这些算子是一些类似乘法和微分的算子的线性组合，当应用于周期函数时，保持周期性。最后，我们构造了$L^2(\mathcal C)$和$l_2(\mathbb Z)$的Riemann算子，使得所有提到的算子都是连续的。
英文摘要：
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.
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