Orthonormal basis

A subset $\{v_1,...,v_k\}$ of a vector space $\mathbf{V}$, with the inner product $\langle,\rangle$, is called orthonormal if $\langle v_i,v_j \rangle=0$ when $i \neq j$. That is, the vectors are mutually perpendicular. Moreover, they are all required to have length one: $\langle v_i,v_i \rangle=1$.

Representation of 1D Signal Using An Orthonormal Basis

Orthonormal basis function

$$\int_{-\infty}^\infty \phi(x,u_1) \phi^*(x,u_2) dx = \begin{cases}1, u_1 = u_2 \\ 0, u_1 \neq u_2\end{cases}$$

• Each basis has norm 1
• Different bases are orthogonal to each other

Inverse transform

• Representing $f(x)$ as integral (limit of sum) of $φ(x,u)$ for all $u$, with weight $F(u)$ $$\int_{-\infty}^\infty F(u) \phi(x,u) \, du$$

Forward transform

• determining the weight through inner product $$F(u) = \langle f(x), \phi(x,u)\rangle = \int_{-\infty}^\infty f(x) \phi^*(x,u) \, dx$$

Orthonormal basis vectors

see Orthonormal basis vectors

References:

1. https://mathworld.wolfram.com/OrthonormalBasis.html
2. https://www.sciencedirect.com/topics/computer-science/orthonormal-basis