Orthogonal system - Definition and Overview

In mathematics, two functions <math>f<math> and <math>g<math> are orthogonal if their inner product <math>\langle f,g\rangle<math> 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

<math> \langle f,g\rangle = \int f^*(x) g(x)\,dx , <math>

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

See also: orthogonal polynomials.