# orthogonal functions

**orthogonal functions** 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

*i*≠

*j*,

*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

*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.

#### More From encyclopedia.com

Function , A function is a mathematical relationship between two sets of real numbers. These sets of numbers are related to each other by a rule that assigns ea… Primitive Recursion , primitive recursive function A function that can be obtained from certain initial functions by a finite number of applications of composition and pri… wavelet , wavelet •mallet, palette, pallet, valet •tablet • pamphlet • aglet • anklet •candlelit • hamlet •Caplet, chaplet •lamplit • flatlet • mantlet •haslet… Domain , Domain
The domain of a relation is the set that contains all the first elements, x, from the ordered pairs (x,y) that make up the relation. In mathem… Equation , equation An expression that asserts the equality of two terms. To be precise, an equation has the following form. Let Σ be a signature and let t1(X1,… Spline , spline / splīn/ • n. 1. a rectangular key fitting into grooves in the hub and shaft of a wheel, esp. one formed integrally with the shaft that allows…

#### You Might Also Like

#### NEARBY TERMS

**orthogonal functions**