orthogonal functions

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orthogonal functions Let f1(x), f2(x),…, fn(x)

be a set of functions defined on the interval (a,b); also let w(x) be a given positive function (a weight function) on (a,b). The functions fi(x), i = 1,2,…,n

are said to be orthogonal with respect to the interval (a,b) and weight function w(x), if ij, i,j = 1,2,…,n

If, for i = j, i = 1,2,…,n

then the functions are said to be orthonormal.

A similar property is defined when (a,b) is replaced by the set of points x1,x2,…,xN

and the integral is replaced by a sum, ij

Orthogonal functions play an important part in the approximation of functions and data.