Even and odd functions

Recall that a function $~$f : \mathbb{R} \to \mathbb{R}$~$ is [even_function even] if $~$f(-x) = f(x)$~$, and [odd_function odd] if $~$f(-x) = - f(x)$~$. A typical example of an even function is $~$f(x) = x^2$~$ or $~$f(x) = \cos x$~$, while a typical example of an odd function is $~$f(x) = x^3$~$ or $~$f(x) = \sin x$~$.

We can think about evenness and oddness in terms of group theory as follows. There is a group called the cyclic group $~$C_2$~$ of order $~$2$~$ acting on the set of functions $~$\mathbb{R} \to \mathbb{R}$~$: in other words, each element of the group describes a function of type

$$~$ (\mathbb{R} \to \mathbb{R}) \to (\mathbb{R} \to \mathbb{R}) $~$$

meaning that it takes as input a function $~$\mathbb{R} \to \mathbb{R}$~$ and returns as output another function $~$\mathbb{R} \to \mathbb{R}$~$.

$~$C_2$~$ has two elements which we'll call $~$1$~$ and $~$-1$~$. $~$1$~$ is the identity element: it acts on functions by sending a function $~$f(x)$~$ to the same function $~$f(x)$~$ again. $~$-1$~$ sends a function $~$f(x)$~$ to the function $~$f(-x)$~$, which visually corresponds to reflecting the graph of $~$f(x)$~$ through the y-axis. The group multiplication is what the names of the group elements suggests, and in particular $~$(-1) \times (-1) = 1$~$, which corresponds to the fact that $~$f(-(-x)) = f(x)$~$.

Any time a group $~$G$~$ acts on a set $~$X$~$, it's interesting to ask what elements are [invariant_under_a_group_action invariant] under that group action. Here the invariants of functions under the action of $~$C_2$~$ above are the even functions, and they form a [subspace] of the vector space of all functions.

It turns out that every function is uniquely the sum of an even and an odd function, as follows:

$$~$f(x) = \underbrace{\frac{f(x) + f(-x)}{2}}_{\text{even}} + \underbrace{\frac{f(x) - f(-x)}{2}}_{\text{odd}}.$~$$

This is a special case of various more general facts in representation theory, and in particular can be thought of as the simplest case of the [discrete_Fourier_transform discrete Fourier transform], which in turn is a [mathematical_toy_model toy model] of the theory of [Fourier_series Fourier series] and the [Fourier_transform Fourier transform].

It's also interesting to observe that the cyclic group $~$C_2$~$ shows up in lots of other places in mathematics as well. For example, it is also the group describing how even and odd numbers add¹ (where even corresponds to $~$1$~$ and odd corresponds to $~$-1$~$); this is the simplest case of modular arithmetic.

¹_{That is: an even plus an even make an even, an odd plus an odd make an even, and an even plus an odd make an odd.}