Integrals and Banach spaces for finite order distributions

Talvila, Erik. "Integrals and Banach spaces for finite order distributions." Czechoslovak Mathematical Journal 62.1 (2012): 77-104. <http://eudml.org/doc/246963>.

@article{Talvila2012,
abstract = {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.},
author = {Talvila, Erik},
journal = {Czechoslovak Mathematical Journal},
keywords = {regulated function; regulated primitive integral; Banach space; Banach lattice; Banach algebra; Schwartz distribution; generalized function; distributional Denjoy integral; continuous primitive integral; Henstock-Kurzweil integral; primitive; regulated function; regulated primitive integral; Banach lattice; distributional Denjoy integral; continuous primitive integral; Henstock-Kurzweil integral},
language = {eng},
number = {1},
pages = {77-104},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Integrals and Banach spaces for finite order distributions},
url = {http://eudml.org/doc/246963},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Talvila, Erik
TI - Integrals and Banach spaces for finite order distributions
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 1
SP - 77
EP - 104
AB - 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.
LA - eng
KW - regulated function; regulated primitive integral; Banach space; Banach lattice; Banach algebra; Schwartz distribution; generalized function; distributional Denjoy integral; continuous primitive integral; Henstock-Kurzweil integral; primitive; regulated function; regulated primitive integral; Banach lattice; distributional Denjoy integral; continuous primitive integral; Henstock-Kurzweil integral
UR - http://eudml.org/doc/246963
ER -