## Quick Reference

Let

*f*_{1}(*x*), *f*_{2}(*x*),…, *f** _{n}*(

*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 *f** _{i}*(

*x*),

*i* = 1,2,…,*n*

are said to be orthogonal with respect to the interval (*a*,*b*) and weight function *w*(*x*), if

∫^{b}_{a}*w*(*x*) *f** _{i}*(

*x*)

*f*

*(*

_{j}*x*) d

*x*= 0,

*i* ≠ *j*, *i*,*j* = 1,2,…,*n*

If, for *i* = *j*,

∫^{a}_{b}*w*(*x*) *f*_{i}^{2}(*x*) d*x* = 1,

*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

*x*_{1},*x*_{2},…,*x*_{n}

and the integral is replaced by a sum,

*i* ≠ *j*

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

*Subjects:*
Computing.

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