nLab
bump function

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

A bump function is a smooth function with compact support, especially one that is not zero on a space that it not compact.

One reason the category SmthMfd of smooth manifolds and smooth functions is important, hence one reason why differential geometry is special, is that a good supply of bump functions exists; this fact accounts for the great flexibility of smooth objects, in stark contrast to analytic geometry or algebraic varieties (since non-zero analytic or algebraic functions with compact support exist only on compact spaces). For instance, bump functions can be used to construct partitions of unity, which can in turn be used to smoothly patch together local structures into a global structure without obstruction, as for example Riemannian metrics and more generally inner products on vector bundles (see this prop.).

Constructions

Definition

A bump function is a function on Cartesian space n\mathbb{R}^n, for some nn \in \mathbb{R} with values in the real numbers \mathbb{R}

b: n b \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}

such that

  1. bb is smooth;

  2. bb has compact support.

Example

For every closed ball B x 0(ϵ)={x n|xx 0ϵ} nB_{x_0}(\epsilon) = \{x \in \mathbb{R}^n \,\vert\, {\Vert x - x_0 \Vert} \leq \epsilon\} \subset \mathbb{R}^n there exists a bump function b: nb \colon \mathbb{R}^n \to \mathbb{R} (def. 1) with

Supp(b)=B x(ϵ). Supp(b) = B_x(\epsilon) \,.
Proof

Consider the function

ϕ: n \phi \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}

given by

ϕ(x){exp(1x 21) | x<1 0 | otherwise. \phi(x) \;\coloneqq\; \left\{ \array{ \exp\left( \frac{1}{{\Vert x \Vert}^2 - 1} \right) & \vert & { \Vert x \Vert} \lt 1 \\ 0 &\vert & \text{otherwise} } \right. \,.

By construction the support of this function is the closed unit ball at the origin, Supp(ϕ)=B 0(1)Supp(\phi) = B_0(1).

We claim that ϕ\phi is smooth. That it is smooth away from rx=0r \coloneqq {\Vert x \Vert} = 0 is clear, hence smoothness only need to be checked at r=0r = 0, where it amounts to demanding that all the derivatives of the exponential function vanish as r0r \to 0.

But that is the case since

ddr(exp(1r 21))=2r(r 21) 2exp(1r 21). \frac{d}{d r} \left( \exp\left( \frac {1} { r^2 - 1 } \right) \right) = \frac{ -2 r } { \left( r^2 - 1 \right)^2 } \exp\left( \frac {1} { r^2 - 1 } \right) \,.

This clearly tends to zero as r1r \to 1. A quick way to see this is to consider the inverse function and expand the exponential to see that this tends to \infty as r1r \to 1:

(1r 2) 22rexp(11r 2)= n=0 1n!(1r 2) 22r1(1r 2) n \frac{ \left( 1- r^2 \right)^2 } { 2 r } \exp\left( \frac {1} { 1- r^2 } \right) = \sum_{n = 0}^\infty \frac{1}{n!} \frac{ \left( 1- r^2 \right)^2 } { 2 r } \frac{1} { (1- r^2)^n }

The form of the higher derivatives is similar but with higher inverse powers of (r 21)(r^2 -1) and so this conclusion remains the same for all derivatives. Hence ϕ\phi is smooth.

Now for arbitrary radii ε>0\varepsilon \gt 0 define

ϕ ε(x)ϕ(x/ε). \phi_\varepsilon(x) \coloneqq \phi(x/\varepsilon) \,.

This is clearly still smooth and Supp(ϕ ε)=B 0(ϵ)Supp(\phi_{\varepsilon}) = B_0(\epsilon).

Finally the function xϕ ε(xx 0)x \mapsto \phi_\varepsilon(x-x_0) has support the closed ball B x 0(ε)B_{x_0}(\varepsilon).

Define

ψ εϕ εϕ ε 1 \psi_\varepsilon \coloneqq \frac{\phi_\varepsilon}{{\|\phi_\varepsilon\|}_1}

so that the family (ψ ε) ε(\psi_\varepsilon)_\varepsilon is a smooth approximation to the identity? (of convolution, the Dirac functional), compactly supported on the closed ball of radius ε\varepsilon and having an L 1L^1-norm equal to 11. Then, it is standard that for any pair KVK \subset V with KK compact and VV open in a cartesian space n\mathbb{R}^n, we can choose an open UU containing KK and with compact closure contained in VV, and then taking the convolution product

ψ ε*χ U\psi_\varepsilon \ast \chi_U

of ψ ε\psi_\varepsilon with the characteristic function χ U\chi_U, for ε\varepsilon sufficiently small, we obtain a smooth function equal to 11 on KK and equal to 00 outside VV.

By performing the above construction in charts, we obtain, in any smooth manifold, a smooth function equal to 11 on any compact subspace KK and equal to 00 outside any neighbourhood VV of KK. (This is a smooth regularity property.)

Properties

Proposition

(Paley-Wiener-Schwartz theorem)

For nn \in \mathbb{N} the vector space C c ( n)C^\infty_c(\mathbb{R}^n) of bump functions on Euclidean space n\mathbb{R}^n is (algebraically and topologically) isomorphic, via the Fourier transform, to the space of entire functions FF on n\mathbb{C}^n which satisfy the following estimate: there is a positive real number BB such that for every integer N>0N \gt 0 there is a real number C NC_N such that:

ξ n(F(ξ)C N(1+|ξ|) Nexp(B|Im(ξ)|)). \underset{\xi \in \mathbb{C}^n}{\forall} \left( {\Vert F(\xi) \Vert} \le C_N (1 + {\vert \xi\vert })^{-N} \exp{ (B \; |Im(\xi)|)} \right) \,.

References

A different example is examined in detail in

  • Juan Arias de Reyna, An infinitely differentiable function with compact support: Definition and properties, arXiv:1702.05442 (English translation of: Definición y estudio de una función indefinidamente diferenciable de soporte compacto, Rev. Real Acad. Ciencias 76 (1982) 21-38 available from Universidad de Sevilla repository)

One can also build bump functions that are nowhere analytic (as opposed to merely on the boundary of the support, as above) using, for instance, the Fabius function:

Revised on September 5, 2017 14:35:21 by Urs Schreiber (77.56.177.247)