Orthogonal functions

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In mathematics, two functions $f$ and $g$ are called 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. Here, the star is the complex conjugate.

For an intuitive perspective on this inner product, suppose approximating vectors $\vec{f}$ and $\vec{g}$ are created whose entries are the values of the functions f and g, sampled at equally spaced points. Then this inner product between f and g can be roughly understood as the dot product between approximating vectors $\vec{f}$ and $\vec{g}$ , in the limit as the number of sampling points goes to infinity. Thus, roughly, two functions are orthogonal if their approximating vectors are perpendicular (under this common inner product).[1]

See also Hilbert space for a more rigorous 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:

[edit] See also

Orthogonal polynomials.

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