# A New Method of Extension of Local Maps of Banach Spaces. Applications and Examples

## TL;DR

This paper introduces blid maps as a new tool to extend local maps in Banach spaces lacking smooth bump functions, enabling advances in local analysis and applications like derivative reconstruction and solving cohomological equations.

## Contribution

It proposes blid maps as an alternative to bump functions for extending local maps in non-smooth Banach spaces, expanding the scope of local analysis techniques.

## Key findings

- Blid maps enable extension of smooth local maps in spaces without bump functions.
- Application to reconstruct maps from derivatives at a point.
- Facilitates finding global solutions to cohomological equations.

## Abstract

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 $C^q[0,1], q=0,1,...$. 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.