Orthogonal functions

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In mathematics, two functions <math>f</math> and <math>g</math> are called 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. Here, the star is the complex conjugate. 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.de:Orthogonal (Funktion)