Orthogonal functions

In mathematics, two functions f and g are called orthogonal if their inner product 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

with appropriate integration boundaries. Here, the star is the complex conjugate.

For an intuitive perspective on this inner product, suppose approximating vectors and 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 and , 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:

• Hermite polynomials
• Legendre polynomials
• Spherical harmonics
• Walsh functions
• Zernike polynomials

See also

• Orthogonal polynomials.

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