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Orthogonal function


In mathematics, orthogonal functions belong to a function space which is a vector space (usually over R) that has 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:

The functions f and g are orthogonal when this integral is zero: As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space.

Suppose {fn}, n = 0, 1, 2, … is a sequence of orthogonal functions. If fn has positive support then is the L2-norm of fn, and the sequence has functions of L2-norm one, forming an orthonormal sequence. The possibility that an integral is unbounded must be avoided, hence attention is restricted to square-integrable functions.


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