TY - GEN

T1 - New Method of Smooth Extension of Local Maps on Linear Topological Spaces. Applications and Examples

AU - Belitskii, Genrich

AU - Rayskin, Victoria

N1 - Publisher Copyright:
© 2020, Springer Nature Switzerland AG.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - The question of extension of locally defined maps to the entire space arises in many problems of analysis (e.g., local linearization of functional equations). A known classical method of extension of smooth local maps on Banach spaces uses smooth bump functions. However, such functions are absent in the majority of infinite-dimensional spaces. We suggest a new approach to localization of Banach spaces with the help of locally identical maps, which we call blid maps. In addition to smooth spaces, blid maps also allow to extend local maps on non-smooth spaces (e.g., Cq[ 0, 1 ], q= 0, 1, 2,.. ). For the spaces possessing blid maps, we show how to reconstruct a map from its derivatives at a point (see the Borel Lemma). We also demonstrate how blid maps assist in finding global solutions of cohomological equations having linear transformation of the argument. We present application of blid maps to local differentiable linearization of maps on Banach spaces. We discuss differentiable localization for metric spaces (e.g., ), prove an extension result for locally defined maps and present examples of such extensions for the specific metric spaces. In conclusion, we formulate open problems.

AB - The question of extension of locally defined maps to the entire space arises in many problems of analysis (e.g., local linearization of functional equations). A known classical method of extension of smooth local maps on Banach spaces uses smooth bump functions. However, such functions are absent in the majority of infinite-dimensional spaces. We suggest a new approach to localization of Banach spaces with the help of locally identical maps, which we call blid maps. In addition to smooth spaces, blid maps also allow to extend local maps on non-smooth spaces (e.g., Cq[ 0, 1 ], q= 0, 1, 2,.. ). For the spaces possessing blid maps, we show how to reconstruct a map from its derivatives at a point (see the Borel Lemma). We also demonstrate how blid maps assist in finding global solutions of cohomological equations having linear transformation of the argument. We present application of blid maps to local differentiable linearization of maps on Banach spaces. We discuss differentiable localization for metric spaces (e.g., ), prove an extension result for locally defined maps and present examples of such extensions for the specific metric spaces. In conclusion, we formulate open problems.

KW - Bump functions

KW - Local maps

KW - Map extensions

UR - http://www.scopus.com/inward/record.url?scp=85101578702&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-60107-2_19

DO - 10.1007/978-3-030-60107-2_19

M3 - Conference contribution

AN - SCOPUS:85101578702

SN - 9783030601065

T3 - Springer Proceedings in Mathematics and Statistics

SP - 353

EP - 368

BT - Progress on Difference Equations and Discrete Dynamical Systems - 25th ICDEA, 2019

A2 - Baigent, Steve

A2 - Bohner, Martin

A2 - Elaydi, Saber

PB - Springer

T2 - 25th International Conference on Difference Equations and Applications, ICDEA 2019

Y2 - 24 June 2019 through 28 June 2019

ER -