Orthogonal functions

In mathematics, two functions $f$ and $g$ are orthogonal if their inner product $\langle f,g\rangle$ is zero. Whether or not two particular functions are orthogonal depends on how their inner product has been defined. A typical definition of an inner product for functions is

$\langle f,g\rangle = \int f^*(x) g(x)\,dx ,$

with appropriate integration boundaries. See also Hilbert space for more background.

Solutions of linear differential equations with boundary conditions can often be written as a weighted sum of orthogonal solution functions (a.k.a. eigenfunctions).

Examples of sets of orthogonal functions:

Hermite polynomials
Legendre polynomials
Spherical harmonics
Walsh functions

Orthogonal functions

See also: Orthogonal functions, Differential equation, Eigenfunction, Hermite polynomials, Hilbert space, Inner product, Legendre polynomials, Mathematics, Orthogonal polynomials, Spherical harmonics