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 ${mathcal {D}}$

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

Statement of the theorem定理的描述

Let $X$

${displaystyle kin {mathcal {D}}'(X imes Y)}$

{displaystyle leftlangle k,uotimes v ight angle =leftlangle Kv,u ight angle }

for every ${displaystyle uin {mathcal {D}}(X),vin {mathcal {D}}(Y)}$

${displaystyle uin {mathcal {D}}(X),vin {mathcal {D}}(Y)}$

Note

Given a distribution ${displaystyle kin {mathcal {D}}'(X imes Y)}$

{displaystyle Kv=int _{Y}k(cdot ,y)v(y)dy}

so that

${displaystyle langle Kv,u angle =int _{X}int _{Y}k(x,y)v(y)u(x)dydx}$ .

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.

External links

G. L. Litvinov (2001) [1994], "Nuclear bilinear form", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

Categories: Generalized functions Transforms Theorems in functional analysis

>> 施瓦兹引理:https://baike.baidu.com/item/施瓦兹引理/18984053

>>Schwartz space:https://en.wikipedia.org/wiki/Schwartz_space

>>Kernel:https://en.wikipedia.org/wiki/Kernel

>>FOURIER STANDARD SPACES and the Kernel Theorem：https://www.univie.ac.at/nuhag-php/dateien/talks/3338_Garching1317.pdf

>>施瓦兹广义函数理论的成因探析：http://www.doc88.com/p-3498616571581.html

>>施瓦兹空间的成因解析：http://www.doc88.com/p-5778688208231.html

>>Cours d'analyse. Théorie des distributions et analyse de Fourier(英文).PDF ：https://max.book118.com/html/2017/0502/103891395.shtm