Linear combinations of partitions of unity with restricted supports

Christian Richter. "Linear combinations of partitions of unity with restricted supports." Studia Mathematica 153.1 (2002): 1-11. <http://eudml.org/doc/285303>.

@article{ChristianRichter2002,
abstract = {Given a locally finite open covering of a normal space X and a Hausdorff topological vector space E, we characterize all continuous functions f: X → E which admit a representation $f = ∑_\{C∈\} a_\{C\}φ_\{C\}$ with $a_\{C\} ∈ E$ and a partition of unity $\{φ_\{C\}: C ∈ \}$ subordinate to . As an application, we determine the class of all functions f ∈ C(||) on the underlying space || of a Euclidean complex such that, for each polytope P ∈ , the restriction $f|_P$ attains its extrema at vertices of P. Finally, a class of extremal functions on the metric space $([-1,1]^\{m\},d_\{∞\})$ is characterized, which appears in approximation by so-called controllable partitions of unity.},
author = {Christian Richter},
journal = {Studia Mathematica},
keywords = {partition of unity subordinate to a covering; continuous selection; polyhedral complex; entropy numbers; nonlinear approximation; normal topological space},
language = {eng},
number = {1},
pages = {1-11},
title = {Linear combinations of partitions of unity with restricted supports},
url = {http://eudml.org/doc/285303},
volume = {153},
year = {2002},
}

TY - JOUR
AU - Christian Richter
TI - Linear combinations of partitions of unity with restricted supports
JO - Studia Mathematica
PY - 2002
VL - 153
IS - 1
SP - 1
EP - 11
AB - Given a locally finite open covering of a normal space X and a Hausdorff topological vector space E, we characterize all continuous functions f: X → E which admit a representation $f = ∑_{C∈} a_{C}φ_{C}$ with $a_{C} ∈ E$ and a partition of unity ${φ_{C}: C ∈ }$ subordinate to . As an application, we determine the class of all functions f ∈ C(||) on the underlying space || of a Euclidean complex such that, for each polytope P ∈ , the restriction $f|_P$ attains its extrema at vertices of P. Finally, a class of extremal functions on the metric space $([-1,1]^{m},d_{∞})$ is characterized, which appears in approximation by so-called controllable partitions of unity.
LA - eng
KW - partition of unity subordinate to a covering; continuous selection; polyhedral complex; entropy numbers; nonlinear approximation; normal topological space
UR - http://eudml.org/doc/285303
ER -