$A$-differentiability and $A$-analyticity
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- by P. M. Gadea and J. Muñoz Masqué PDF
- Proc. Amer. Math. Soc. 124 (1996), 1437-1443 Request permission
Abstract:
Let $A$ be a finite-dimensional commutative algebra over $\mathbb {R}$ and let $C_{A}^{r}(U)$, $C^{\omega }(U,A)$ and $\mathcal { O}_{A}(U)$ be the ring of $A$-differentiable functions of class $C^{r}, 0 \leq r \leq \infty$, the ring of real analytic mappings with values in $A$ and the ring of $A$-analytic functions, respectively, defined on an open subset $U$ of $A^{n}$. We prove two basic results concerning $A$-differentiability and $A$-analyticity: $1^{st}$) $\mathcal { O}_{A}(U) = C^{\infty }_{A}(U) \bigcap C^{\omega }(U,A)$, $2^{nd}$) $\mathcal { O}_{A}(U) = C^{\infty }_{A}(U)$ if and only if $A$ is defined over $\mathbb {C}$.References
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Additional Information
- P. M. Gadea
- Affiliation: Instituto de Matemáticas y Física Fundamental Consejo Superior de Investigaciones Científicas Serrano 123, 28006-Madrid, Spain
- Email: pmgadea@gugu.usal.es
- J. Muñoz Masqué
- Affiliation: Instituto de Electrónica de Comunicaciones Consejo Superior de Investigaciones Científicas Serrano 144, 28006-Madrid, Spain
- Email: vctqu01@cc.csic.es
- Received by editor(s): March 1, 1994
- Received by editor(s) in revised form: September 16, 1994
- Additional Notes: Supported by DGICYT (Spain) grant no. PB89-0004
- Communicated by: Albert Baernstein II
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 1437-1443
- MSC (1991): Primary 30G35; Secondary 26E05, 26E10, 16P10
- DOI: https://doi.org/10.1090/S0002-9939-96-03070-5
- MathSciNet review: 1301495