We introduce the notion of quasi-inner function and show that the product $u={\rho }_{\infty }\prod {\rho }_v$ of $m+1$ ratios of local L-factors ${\rho }_v(z)={\gamma }_v(z)/{\gamma }_v(1-z)$ over a finite set F of places of $\mathbb{Q}$ inclusive of the archimedean place is quasi-inner on the left of the critical line $\Re (z)=\frac{1}{2}$ in the following sense. The off diagonal part $u_{21}$ of the matrix of the multiplication by u in the orthogonal decomposition of the Hilbert space $L^2$ of square integrable functions on the critical line into the Hardy space $H^2$ and its orthogonal complement is a compact operator. When interpreted on the unit disk, the quasi-inner condition means that the associated Haenkel matrix is compact. We show that none of the individual non-archimedean ratios ${\rho }_v$ is quasi-inner and, in order to prove our main result we use Gauss multiplication theorem to factor the archimedean ratio ${\rho }_{\infty }$ into a product of m quasi-inner functions whose product with each ${\rho }_v$ retains the property to be quasi-inner. Finally we prove that Sonin's space is simply the kernel of the diagonal part $u_{22}$ for the quasi-inner function $u={\rho }_{\infty }$, and when $u(F)={\prod }_{v\in F}{\rho }_v$ the kernels of the $u{(F)}_{22}$ form an inductive system of infinite dimensional spaces which are the semi-local analogues of (classical) Sonin's spaces.

我们介绍了拟内函数的概念，并证明了该乘积$ü={\rho }_{\infty }\prod {\rho }_v$ 的 $米+\mathrm{1个}$局部L因子之比${\rho }_v（ž）={\gamma }_v（ž）/{\gamma }_v（\mathrm{1个}-ž）$在以下位置的有限集F上$\mathbb{问}$ 包括阿基米德人的地方在内，在临界线的左侧是准内部 $\Re （ž）=\frac{\mathrm{1个}}{\mathrm{2个}}$在以下意义上。离对角线部分${ü}_{\mathrm{21岁}}$相乘的矩阵由ü在Hilbert空间的正交分解${\mathrm{大号}}^{\mathrm{2个}}$ 进入Hardy空间的关键线上的平方可积函数 $H^{\mathrm{2个}}$它的正交补码是一个紧凑算子。当在单位圆盘上解释时，准内部条件意味着关联的汉克尔矩阵是紧凑的。我们表明，没有任何非原始档案比率${\rho }_v$ 是准内部的，为了证明我们的主要结果，我们使用高斯乘法定理来分解阿基米德比率 ${\rho }_{\infty }$分为m个拟内部函数的乘积，每个乘积的乘积${\rho }_v$保留该财产为准内部财产。最后，我们证明Sonin的空间只是对角线部分的核心${ü}_{22}$ 用于准内部函数 $ü={\rho }_{\infty }$， 什么时候 $ü（F）={\prod }_{v\in F}{\rho }_v$ 的内核 $ü{（F）}_{22}$ 形成一个无限维空间的归纳系统，它是（经典）索宁空间的半局部类似物。