| Title:
|
Estimates of the remainder in Taylor’s theorem using the Henstock-Kurzweil integral (English) |
| Author:
|
Talvila, Erik |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
55 |
| Issue:
|
4 |
| Year:
|
2005 |
| Pages:
|
933-940 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
When a real-valued function of one variable is approximated by its $n$th degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue $p$-norms in cases where $f^{(n)}$ or $f^{(n+1)}$ are Henstock-Kurzweil integrable. When the only assumption is that $f^{(n)}~$ is Henstock-Kurzweil integrable then a modified form of the $n$th degree Taylor polynomial is used. When the only assumption is that $f^{(n)}\in C^0$ then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1. (English) |
| Keyword:
|
Taylor’s theorem |
| Keyword:
|
Henstock-Kurzweil integral |
| Keyword:
|
Alexiewicz norm |
| MSC:
|
26A24 |
| MSC:
|
26A39 |
| idZBL:
|
Zbl 1081.26002 |
| idMR:
|
MR2184374 |
| . |
| Date available:
|
2009-09-24T11:29:07Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128035 |
| . |
| Reference:
|
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| Reference:
|
[2] V. G. Čelidze and A. G. Džvaršeǐšvili: The Theory of the Denjoy Integral and Some Applications.World Scientific, Singapore, 1989. MR 1036270 |
| Reference:
|
[3] G. B. Folland: Remainder estimates in Taylor’s theorem.Amer. Math. Monthly 97 (1990), 233–235. Zbl 0737.41031, MR 1048439, 10.2307/2324693 |
| Reference:
|
[4] S. Saks: Theory of the Integral.Monografie Matematyczne, Warsaw, 1937. Zbl 0017.30004 |
| Reference:
|
[5] C. Swartz: Introduction to Gauge Integrals.World Scientific, Singapore, 2001. Zbl 0982.26006, MR 1845270 |
| Reference:
|
[6] H. B. Thompson: Taylor’s theorem using the generalized Riemann integral.Amer. Math. Monthly 96 (1989), 346–350. Zbl 0682.26001, MR 0992083, 10.2307/2324092 |
| Reference:
|
[7] R. Výborný: Some applications of Kurzweil-Henstock integration.Math. Bohem. 118 (1993), 425–441. MR 1251885 |
| Reference:
|
[8] W. H. Young: The Fundamental Theorems of the Differential Calculus.Cambridge University Press, Cambridge, 1910. |
| . |
