| Title:
|
On extending ${\rm C}^{k}$ functions from an open set to $\mathbb R$ with applications (English) |
| Author:
|
Burgess, Walter D. |
| Author:
|
Raphael, Robert M. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
73 |
| Issue:
|
2 |
| Year:
|
2023 |
| Pages:
|
487-498 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For $k\in {\mathbb N} \cup \{\infty \}$ and $U$ open in $ {\mathbb R}$, let ${\rm C}^{k}(U)$ be the ring of real valued functions on $U$ with the first $k$ derivatives continuous. It is shown that for $f\in {\rm C}^{k}(U)$ there is $g\in {\rm C}^{\infty } ({\mathbb R})$ with $U\subseteq {\rm coz} g$ and $h\in {\rm C}^{k} ({\mathbb R})$ with $fg|_U=h|_U$. The function $f$ and its $k$ derivatives are not assumed to be bounded on $U$. The function $g$ is constructed using splines based on the Mollifier function. Some consequences about the ring ${\rm C}^{k} ({\mathbb R})$ are deduced from this, in particular that ${\rm Q}_{\rm cl} ({\rm C}^{k} ({\mathbb R})) = {\rm Q}({\rm C}^{k} ({\mathbb R}))$. (English) |
| Keyword:
|
${\rm C}^k$ function |
| Keyword:
|
spline |
| Keyword:
|
ring of quotient |
| Keyword:
|
Mollifier function |
| MSC:
|
13B30 |
| MSC:
|
26A24 |
| MSC:
|
54C30 |
| idZBL:
|
Zbl 07729519 |
| idMR:
|
MR4586906 |
| DOI:
|
10.21136/CMJ.2023.0445-21 |
| . |
| Date available:
|
2023-05-04T17:46:43Z |
| Last updated:
|
2025-07-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151669 |
| . |
| Reference:
|
[1] Anderson, D. F., Badawi, A.: Divisibility conditions in commutative rings with zerodivisors.Commun. Algebra 30 (2002), 4031-4047. Zbl 1063.13003, MR 1922326, 10.1081/AGB-120005834 |
| Reference:
|
[2] Azarpanah, F., Ghashghaei, E., Ghoulipour, M.: $C(X)$: Something old and something new.Commun. Algebra 49 (2021), 185-206. Zbl 1453.13008, MR 4193623, 10.1080/00927872.2020.1797070 |
| Reference:
|
[3] Barr, M., Burgess, W. D., Raphael, R.: Ring epimorphisms and $C(X)$.Theory Appl. Categ. 11 (2003), 283-308. Zbl 1042.54007, MR 1988400 |
| Reference:
|
[4] Barr, M., Kennison, J. F., Raphael, R.: Limit closures of classes of commutative rings.Theory Appl. Categ. 30 (2015), 229-304. Zbl 1327.13084, MR 3322157 |
| Reference:
|
[5] Blair, R. L., Hager, A. W.: Extensions of zero-sets and real-valued functions.Math. Z. 136 (1974), 41-52. Zbl 0264.54011, MR 0385793, 10.1007/BF01189255 |
| Reference:
|
[6] Fine, N. J., Gillman, L., Lambek, J.: Rings of Quotients of Rings of Functions.McGill University Press, Montreal (1966). Zbl 0143.35704, MR 0200747 |
| Reference:
|
[7] Gillman, L., Jerison, M.: Ring of Continuous Functions.Graduate Texts in Mathematics 43. Springer, New York (1976). Zbl 0327.46040, MR 0407579, 10.1007/978-1-4615-7819-2 |
| Reference:
|
[8] Henriksen, M.: Rings of continuous functions from an algebraic point of view.Ordered Algebraic Structures Mathematics and its Applications 55. Kluwer Academic, Dordrecht (1989), 143-174. Zbl 0739.46011, MR 1094833, 10.1007/978-94-009-2472-7_12 |
| Reference:
|
[9] Hörmander, L.: The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis.Grundlehren der Mathematischen Wissenschaften 256. Springer, Berlin (1983). Zbl 0521.35001, MR 0717035, 10.1007/978-3-642-96750-4 |
| Reference:
|
[10] Knox, M. L., Levy, R., McGovern, W. W., Shapiro, J.: Generalizations of complemented rings with applications to rings of functions.J. Algebra Appl. 8 (2009), 17-40. Zbl 1173.13002, MR 2191531, 10.1142/S0219498809003138 |
| Reference:
|
[11] Lambek, J.: Lectures on Rings and Modules.Chelsea Publishing, New York (1976). Zbl 0365.16001, MR 0419493 |
| Reference:
|
[12] Stenström, B.: Rings of Quotients: An Introduction to Methods of Ring Theory.Die Grundlehren der mathematischen Wissenschaften 217. Springer, Berlin (1975). Zbl 0296.16001, MR 0389953, 10.1007/978-3-642-66066-5 |
| . |
