Integrals and Banach spaces for finite order distributions

Title:

Integrals and Banach spaces for finite order distributions

Creator:

Talvila, Erik

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:ab4c33e4-8d62-492a-ac50-20df5eb1aee5
uuid:ab4c33e4-8d62-492a-ac50-20df5eb1aee5

Subject:

regulated function, regulated primitive integral, Banach space, Banach lattice, Banach algebra, Schwartz distribution, generalized function, distributional Denjoy integral, continuous primitive integral, Henstock-Kurzweil integral, and primitive

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

Let $\mathcal B_c$ denote the real-valued functions continuous on the extended real line and vanishing at $-\infty $. Let $\mathcal B_r$ denote the functions that are left continuous, have a right limit at each point and vanish at $-\infty $. Define $\mathcal A^n_c$ to be the space of tempered distributions that are the $n$th distributional derivative of a unique function in $\mathcal B_c$. Similarly with $\mathcal A^n_r$ from $\mathcal B_r$. A type of integral is defined on distributions in $\mathcal A^n_c$ and $\mathcal A^n_r$. The multipliers are iterated integrals of functions of bounded variation. For each $n\in \mathbb N$, the spaces $\mathcal A^n_c$ and $\mathcal A^n_r$ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to $\mathcal B_c$ and $\mathcal B_r$, respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space $\mathcal A_c^1$ is the completion of the $L^1$ functions in the Alexiewicz norm. The space $\mathcal A_r^1$ contains all finite signed Borel measures. Many of the usual properties of integrals hold: Hölder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

77-104

Source:

Czechoslovak Mathematical Journal | 2012 Volume:62 | Number:1

Harvested from:

CDK

Metadata only:

false