| Title:
|
Integrals and Banach spaces for finite order distributions (English) |
| Author:
|
Talvila, Erik |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
77-104 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
regulated function |
| Keyword:
|
regulated primitive integral |
| Keyword:
|
Banach space |
| Keyword:
|
Banach lattice |
| Keyword:
|
Banach algebra |
| Keyword:
|
Schwartz distribution |
| Keyword:
|
generalized function |
| Keyword:
|
distributional Denjoy integral |
| Keyword:
|
continuous primitive integral |
| Keyword:
|
Henstock-Kurzweil integral |
| Keyword:
|
primitive |
| MSC:
|
26A39 |
| MSC:
|
46B42 |
| MSC:
|
46E15 |
| MSC:
|
46F10 |
| MSC:
|
46G12 |
| MSC:
|
46J10 |
| idZBL:
|
Zbl 1249.26012 |
| idMR:
|
MR2899736 |
| DOI:
|
10.1007/s10587-012-0018-5 |
| . |
| Date available:
|
2012-03-05T07:13:47Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142042 |
| . |
| Reference:
|
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