Spaces of polynomial functions of bounded degrees on an embedded manifold and their duals

Shuzo Izumi. "Spaces of polynomial functions of bounded degrees on an embedded manifold and their duals." Annales Polonici Mathematici 113.1 (2015): 1-42. <http://eudml.org/doc/286664>.

@article{ShuzoIzumi2015,
abstract = {Let (U) denote the algebra of holomorphic functions on an open subset U ⊂ ℂⁿ and Z ⊂ (U) its finite-dimensional vector subspace. By the theory of least spaces of de Boor and Ron, there exists a projection $_\{b\}$ from the local ring $_\{n,b\}$ onto the space $Z_\{b\}$ of germs of elements of Z at b. At a general point b ∈ U its kernel is an ideal and $_\{b\}$ induces the structure of an Artinian algebra on $Z_\{b\}$. In particular, this holds at points where the kth jets of elements of Z form a vector bundle for each k ∈ ℕ. For an embedded manifold $X ⊂ ℂ^\{m\}$, we introduce a space of higher order tangents following Bos and Calvi. In the case of curve, using $_\{b\}$, we define the Taylor projector of order d at a general point a ∈ X, generalising results of Bos and Calvi. It is a retraction of $_\{X,a\}$ onto the set of polynomial functions on $X_\{a\}$ of degree up to d. Using the ideal property stated above, we show that the transcendency index, defined by the author, of the embedding of a manifold $X ⊂ ℂ^\{m\}$ is not very high at a general point of X.},
author = {Shuzo Izumi},
journal = {Annales Polonici Mathematici},
keywords = {algebra of holomorphic functions; local ring ; Taylor projector},
language = {eng},
number = {1},
pages = {1-42},
title = {Spaces of polynomial functions of bounded degrees on an embedded manifold and their duals},
url = {http://eudml.org/doc/286664},
volume = {113},
year = {2015},
}

TY - JOUR
AU - Shuzo Izumi
TI - Spaces of polynomial functions of bounded degrees on an embedded manifold and their duals
JO - Annales Polonici Mathematici
PY - 2015
VL - 113
IS - 1
SP - 1
EP - 42
AB - Let (U) denote the algebra of holomorphic functions on an open subset U ⊂ ℂⁿ and Z ⊂ (U) its finite-dimensional vector subspace. By the theory of least spaces of de Boor and Ron, there exists a projection $_{b}$ from the local ring $_{n,b}$ onto the space $Z_{b}$ of germs of elements of Z at b. At a general point b ∈ U its kernel is an ideal and $_{b}$ induces the structure of an Artinian algebra on $Z_{b}$. In particular, this holds at points where the kth jets of elements of Z form a vector bundle for each k ∈ ℕ. For an embedded manifold $X ⊂ ℂ^{m}$, we introduce a space of higher order tangents following Bos and Calvi. In the case of curve, using $_{b}$, we define the Taylor projector of order d at a general point a ∈ X, generalising results of Bos and Calvi. It is a retraction of $_{X,a}$ onto the set of polynomial functions on $X_{a}$ of degree up to d. Using the ideal property stated above, we show that the transcendency index, defined by the author, of the embedding of a manifold $X ⊂ ℂ^{m}$ is not very high at a general point of X.
LA - eng
KW - algebra of holomorphic functions; local ring ; Taylor projector
UR - http://eudml.org/doc/286664
ER -