【摘要】 L²(e^-x2dx)空间的函数在R上一般是不可积的,其Fourier变换是通过Parseval公式（■）=（f,g）作为线性泛函来定义的。本文按L²(dx^?)=L²（R）中函数Fourier变换的定义方法,用极限式形来定义L²(e^-x2dx)中函数的Fourier变换,并由此获得下列结果: 1.L²(e^-x2dx)(记为L_ii)中函数的Fourier变换全体也构成一个Hilbert空间L_Ⅱ。 2.L_ii和L_Ⅱ通过Fourier变换等距同构,因而Fourier变换算子是酉算子。 3.每个Lp（R）（P≧1）空间都是L_ii或者L_Ⅱ的稠密子空间。更多还原

【Abstract】 Generally the functions in the space L² (e^-x2)dx) are not integrable in R, their Fourier transforms are defined by the Parseval formula （f, g） = （f, g） as a linear functional. But in this paper I define it as a limit in the space L²(e^-x2dx), just like the Fourier transforms of the functions in the space L²（dx） =L²（R）,and from this 1 obtained1. All the Fourier tiansforms of the fnnctions in L²(e^-x2dx) (denoted as L_ii) also form a Hilbert space L_Ⅱ.2. Because the space L_ii and L_Ⅱ are isometric through the Fourier transform, the Fourier transform is a unitary operator.3. Every space L^p （R） （P≥1 ） is a dense subspace of L_ii or L_Ⅱ.更多还原