Schwartz kernel theorem施瓦兹核定理

In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz (Schwartz distributions) have a two-variable theory that includes all reasonable bilinear forms on the space of test functions. The space itself consists of smooth functions of compact support.

在数学中，施瓦茨核定理是广义函数理论的一个基本结果，由Laurent Schwartz在1952年发表。广义地说，它表明，由Schwartz引入的广义函数具有双变量理论，包含在测试函数的空间D上的所有合理的双线性形式。空间D自身由紧凑支持型的光滑函数组成。

Statement of the theorem定理的描述

Let and be open sets in . Every distribution defines a continuous linear map such that

让X和Y为Rⁿ上的开放集合。每一个分布定义了一个连续的线性映射从而使得

for every . Conversely, for every such continuous linear map there exists one and only one distribution such that (1) holds. The distribution is the kernel of the map .

对于每一个。相反地，对于每一个这样的连续线性映射K，存在有且仅有一个分布使得(1)成立。分布k就是映射K的核。

Note

Given a distribution one can always write the linear map K informally as

so that

Integral kernels

The traditional kernel functions K(x, y) of two variables of the theory of integral operators having been expanded in scope to include their generalized function analogues, which are allowed to be more singular in a serious way, a large class of operators from D to its dual space D′ of distributions can be constructed. The point of the theorem is to assert that the extended class of operators can be characterised abstractly, as containing all operators subject to a minimum continuity condition. A bilinear form on D arises by pairing the image distribution with a test function.

A simple example is that the identity operator I corresponds to δ(x − y), in terms of the Dirac delta function δ. While this is at most an observation, it shows how the distribution theory adds to the scope. Integral operators are not so 'singular'; another way to put it is that for K a continuous kernel, only compact operators are created on a space such as the continuous functions on [0,1]. The operator I is far from compact, and its kernel is intuitively speaking approximated by functions on [0,1] × [0,1] with a spike along the diagonal x = y and vanishing elsewhere.

This result implies that the formation of distributions has a major property of 'closure' within the traditional domain of functional analysis. It was interpreted (comment of Jean Dieudonné) as a strong verification of the suitability of the Schwartz theory of distributions to mathematical analysis more widely seen. In his Éléments d'analyse volume 7, p. 3 he notes that the theorem includes differential operators on the same footing as integral operators, and concludes that it is perhaps the most important modern result of functional analysis. He goes on immediately to qualify that statement, saying that the setting is too 'vast' for differential operators, because of the property of monotonicity with respect to the support of a function, which is evident for differentiation. Even monotonicity with respect to singular support is not characteristic of the general case; its consideration leads in the direction of the contemporary theory of pseudo-differential operators.

Smooth manifolds

Dieudonné proves a version of the Schwartz result valid for smooth manifolds, and additional supporting results, in sections 23.9 to 23.12 of that book.

References

Hörmander, L. (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., 256, Springer, doi:10.1007/978-3-642-96750-4, ISBN 3-540-12104-8, MR 0717035.