"Taylor Series" analog for functionals?

For a function $f(x)$, it is possible to write it as a taylor series centered around a point $x=a$:

$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a){(x-a)}^{n}}{n!}=f(a)+f'(a)(x-a)+\frac{f''(a)(x-a)^{2}}{2}+...$$

(Of course, there's a lot more mathematical nuance to Taylor Series expansion, I just want to lay it out loosely here as a basis for my intuition.)

I'm wondering if there's anyway to apply this to functionals, that is, a functional $F[f(x)]$ that maps the function $f(x)$ to an output. Is there a way to "rewrite" a functional as a series such as this:

$$F[f(x)]=a_0+a_1(f(x)-\phi(x))+a_2(f(x)-\phi(x))^2+...$$

Where $\phi(x)$ is a function that acts analogously to the point $x=a$ in a Taylor expansion.

(Again, I'm using all of my terminology and notation pretty loosely here. I'm not going for robust mathematical rigoorousness; I just want to express my intuition behind this idea.)

Is this "Functional expanded as a series" idea a thing? What is it called? Does it have any applications?

Solution 1:

Indeed there is. This is used in calculus of variation. Commonly up to and including order two. See https://en.wikipedia.org/wiki/Functional_derivative

Solution 2:

It seems counter intuitive, but if you forget functionals (That is functions from functions to numbers) and just look at functions from functions to functions) you can build a taylor series like idea somewhat intuitively. I'm still working out the details of it, but i have been able to use it to show for example

$$ f(x+1) = f + f' + \frac{1}{2!}f''+\frac{1}{3!}f''' + ... $$

See here: How to build taylor series for infinite dimensional objects? for how to construct such series and for a list of problems that I encounter.

Somehow, most ideas in calculus generalize very cleanly in the context of "functions of functions" but fail to generalize EASILY (not saying they dont at all) in the context of functionals.