What uniquely characterizes the germ of a smooth function?

[lugita15]

Let $X$ be the set of all functions infinitely differentiable at $0$. Let's define an equivalence relation on $X$ by saying that $f\sim g$ if there exists a sufficiently small open interval $I$ containing $0$ such that $f(x)=g(x)$ for all $x$ in $I$. Then the set of germs of $X$ denotes the set $Y$ of equivalence classes of elements of $X$ under this equivalence relation.

My question, what uniquely characterizes the germ of a smooth function? That is to say, what is the minimum information needed to unambiguously specify a single element of $Y$ as opposed to all other elements of $Y$? The nth derivatives of $f$ for all $n$ isn’t enough information, because the function f defined by $f(x)=e^{-\frac{1}{x^2}}$ when $x$ does not equal $0$ and $f(0)=0$ has the same nth derivatives as the function $g(x)=0$ for all $n$, but they still don’t belong to the same germ.

 

[mathman]

Analytic functions form a subset of $X$. Being analytic will meet your requirement, but I don't know if it is the minimum information.

 

[fresh_42]

In your example there is no such interval $I$, so shouldn't it simply be the existence of that open neighborhood which qualifies a germ.

 

[andrewkirk]

Analytic functions form a subset of $X$. Being analytic will meet your requirement, but I don't know if it is the minimum information.

It's not the minimum information, but it provides a great way to deconstruct the search, by taking the quotient of the space of functions that are smooth in a nbd of X over the space of functions that are analytic at X. That way, we need only consider germ-equivalence classes of smooth non-analytic functions.

Given a function $f$ that is analytic at X, the following family of functions gives a different germ at X for every real $a$ and $b$:

$$g(x) = f(x) + a h(x-X)$$
if $x\ge X$, and
$$g(x) = f(x) + b h(X-x)$$
otherwise, where:
$$h(x) = \exp\left(\frac1{x^2}\right)$$
if $x>0$, and
$$h(x)=0$$
otherwise.

So the quotient space has dimension at least 2 as a real vector space. But we know it must be more than that because at least one smooth, non-analytic function that is not in the subspace of the quotient that is thus generated is this function, which is smooth but nowhere real analytic.

 

[mathwonk]

i suspect what mathman meant was that the sequence of derivatives of all orders at 0, is the minimum amount of identifying information of a germ, for functions that are analytic, and presumably only for those. In general I do not believe there is any minimal identifier. I.e. the only data that will determine a germ is the values of the function on some interval, but the values on any smaller interval will also work, so no interval is minimal.

well ok, since the function is continuous, knowing its values on any dense subset of an interval will do. again, not only is the interval not minimal, the dense subset isn't either. so i do not know of any useful solution to this problem.

in the analytic case of course the germs are in one one correspondence with the convergent power series at zero. and in the algebraic case, the germs are in one one correspondence with the "local ring" at the origin, i.e. quotients of polynomials, with bottoms having a non zero constant term.but since the question to describing the quotient space of all germs, maybe there is a way, and maybe andrewkirk's idea to also throw out the known subspace of analytic stuff may help, but i don't see how to describe it.

it may be illuminating to read the sections in courant's calculus referred to as "order of vanishing" in his index. there he discusses the complexity of the related problem of comparing how fast various functions approach zero or infinity as x-->0 or infinity. essentially there is no way to assign a number to this even if we use all real numbers. in particular there are functions that approach infinity faster than x but slower than x^(1+e) for all e > 0. thus no real number exists that measures the speed of increase of such functions. (courant, vol. 1, p. 193). similarly there are functions that approach zero more slowly and others that approach more rapidly than any power of x.

 

[lugita15]

it may be illuminating to read the sections in courant's calculus referred to as "order of vanishing" in his index. there he discusses the complexity of the related problem of comparing how fast various functions approach zero or infinity as x-->0 or infinity. essentially there is no way to assign a number to this even if we use all real numbers. in particular there are functions that approach infinity faster than x but slower than x^(1+e) for all e > 0. thus no real number exists that measures the speed of increase of such functions. (courant, vol. 1, p. 193). similarly there are functions that approach zero more slowly and others that approach more rapidly than any power of x.

Yes, if we define a partial order on the set of all functions infinitely differentiable at 0 by saying that f<g if the limit of f(x)/g(x) as x goes to 0 = 0, and then take a maximal totally ordered subset of that partially ordered set, then the order type of that set will be much bigger than the order type of the real numbers. It will basically be order-isomorphic to the set of all surreal numbers with birthday less than omega_1 (the first uncountable ordinal). See here.