Geodesic
Estimates of the remainder in Taylor’s theorem using the Henstock-Kurzweil integral
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 4, pp. 933-940
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 26A24, 26A39
Keywords: Taylor’s theorem; Henstock-Kurzweil integral; Alexiewicz norm 
@article{CMJ_2005_55_4_a9,
     author = {Talvila, Erik},
     title = {Estimates of the remainder in {Taylor{\textquoteright}s} theorem using the {Henstock-Kurzweil} integral},
     journal = {Czechoslovak Mathematical Journal},
     pages = {933--940},
     year = {2005},
     volume = {55},
     number = {4},
     mrnumber = {2184374},
     zbl = {1081.26002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2005_55_4_a9/}
}
TY  - JOUR
AU  - Talvila, Erik
TI  - Estimates of the remainder in Taylor’s theorem using the Henstock-Kurzweil integral
JO  - Czechoslovak Mathematical Journal
PY  - 2005
SP  - 933
EP  - 940
VL  - 55
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CMJ_2005_55_4_a9/
LA  - en
ID  - CMJ_2005_55_4_a9
ER  - 
%0 Journal Article
%A Talvila, Erik
%T Estimates of the remainder in Taylor’s theorem using the Henstock-Kurzweil integral
%J Czechoslovak Mathematical Journal
%D 2005
%P 933-940
%V 55
%N 4
%U http://geodesic.mathdoc.fr/item/CMJ_2005_55_4_a9/
%G en
%F CMJ_2005_55_4_a9
Talvila, Erik. Estimates of the remainder in Taylor’s theorem using the Henstock-Kurzweil integral. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 4, pp. 933-940. http://geodesic.mathdoc.fr/item/CMJ_2005_55_4_a9/

[1] G. A.  Anastassiou and S. S.  Dragomir: On some estimates of the remainder in Taylor’s formula. J.  Math. Anal. Appl. 263 (2001), 246–263. | DOI | MR

[2] V. G.  Čelidze and A. G.  Džvaršeǐšvili: The Theory of the Denjoy Integral and Some Applications. World Scientific, Singapore, 1989. | MR

[3] G. B.  Folland: Remainder estimates in Taylor’s theorem. Amer. Math. Monthly 97 (1990), 233–235. | DOI | MR | Zbl

[4] S.  Saks: Theory of the Integral. Monografie Matematyczne, Warsaw, 1937. | Zbl

[5] C.  Swartz: Introduction to Gauge Integrals. World Scientific, Singapore, 2001. | MR | Zbl

[6] H. B.  Thompson: Taylor’s theorem using the generalized Riemann integral. Amer. Math. Monthly 96 (1989), 346–350. | DOI | MR | Zbl

[7] R.  Výborný: Some applications of Kurzweil-Henstock integration. Math. Bohem. 118 (1993), 425–441. | MR

[8] W. H.  Young: The Fundamental Theorems of the Differential Calculus. Cambridge University Press, Cambridge, 1910.