Inner Product of Functions

Formal

(Inner Product of Functions)

The inner product of two functions $u : \mathbb{R} \rightarrow \mathbb{R}$ and $v:\mathbb{R} \rightarrow \mathbb{R}$ can be defined as integral

$$\begin{align} \langle u,v\rangle \coloneqq \int_{a}^b u(x)v(x)\,dx \tag{1} \end{align}$$

for lower and upper limits $a,b < \infty$, respectively. if (1) evaluates to 0, the functions $u$ and $v$ are orthogonal. Notice the generalization of inner product as defined for finite-dimensional vectors.

Intuition

The inner product of two functions f(x) and g(x) is a measure of their overall overlap or correlation. If both functions are large and positive in the same regions, their inner product will be a large positive number. If they have opposite signs in the same regions, their inner product will be a large negative number. If their positive and negative parts cancel each other out, their inner product will be close to zero, suggesting they are “uncorrelated” or orthogonal.

Connections

• Forward Links:
• Backward Links: Inner Products