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mathematical analysis > functional analysis > function space > orthogonal function

set theory > function space > orthogonal function

algebra > linear algebra > function space > orthogonal function

category theory > function space > orthogonal function

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orthogonal function

In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval:

${\\displaystyle \\langle f,g\ angle =\\int {\\overline {f(x)}}g(x)\\,dx.}$

The functions ${\\displaystyle f}$ and ${\\displaystyle g}$ are orthogonal when this integral is zero, i.e. ${\\displaystyle \\langle f,\\,g\ angle =0}$ whenever ${\\displaystyle f\ eq g}$ . As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the above integral is the equivalent of a vector dot product; two vectors are mutually independent (orthogonal) if their dot-product is zero.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Orthogonal_functions)

function space

fonction orthogonale

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