TY - CHAP

T1 - A new method of extension of local maps of banach spaces. Applications and examples

AU - Belitskii, Genrich

AU - Rayskin, Victoria

N1 - Publisher Copyright:
©2019 American Mathematical Society.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - A known classical method of extension of smooth local maps of Banach spaces uses smooth bump functions. However, such functions are absent in the majority of infinite-dimensional Banach spaces. This is an obstacle in the development of local analysis, in particular in the questions of extending local maps onto the whole space. We suggest an approach that substitutes bump functions with special maps, which we call blid maps. It allows us to extend smooth local maps from non-smooth spaces, such as Cq [0, 1], q = 0, 1, …. As an example of applications, we show how to reconstruct a map from its derivatives at a point, for spaces possessing blid maps. We also show how blid maps can assist in finding global solutions to cohomological equations having linear transformation of argument.

AB - A known classical method of extension of smooth local maps of Banach spaces uses smooth bump functions. However, such functions are absent in the majority of infinite-dimensional Banach spaces. This is an obstacle in the development of local analysis, in particular in the questions of extending local maps onto the whole space. We suggest an approach that substitutes bump functions with special maps, which we call blid maps. It allows us to extend smooth local maps from non-smooth spaces, such as Cq [0, 1], q = 0, 1, …. As an example of applications, we show how to reconstruct a map from its derivatives at a point, for spaces possessing blid maps. We also show how blid maps can assist in finding global solutions to cohomological equations having linear transformation of argument.

UR - http://www.scopus.com/inward/record.url?scp=85078800346&partnerID=8YFLogxK

U2 - 10.1090/conm/733/14733

DO - 10.1090/conm/733/14733

M3 - Chapter

AN - SCOPUS:85078800346

T3 - Contemporary Mathematics

SP - 61

EP - 72

BT - Contemporary Mathematics

PB - American Mathematical Society

ER -