Explore: Schwartz Distributions

Distributions, also known as Schwartz distributions are a kind of generalized function in mathematical analysis. Distributions make it possible to differentiate

{S}}} , that is, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function. Schwartz space is named after French

were suggested. The problem of multiplication of distributions, a limitation of the Schwartz distribution theory, becomes serious for non-linear problems

Congress of Mathematicians, Schwartz was a plenary speaker and was awarded the Fields Medal for his work on distributions. He was the first French mathematician

containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau

broad terms, that the generalized functions introduced by Schwartz (Schwartz distributions) have a two-variable theory that includes all reasonable bilinear

test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are

tempered distributions S ′ ( R n ) {\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})} by duality of the Fourier transform on the space of Schwartz functions

corresponding distributions. An example of a nuclear countably Hilbert space Φ {\displaystyle \Phi } and its dual Φ ∗ {\displaystyle \Phi ^{*}} is the Schwartz space

and the limit must be understood as taking place in the space of Schwartz distributions. The function φ {\displaystyle \varphi } may also satisfy further