Orthogonal functions: Difference between revisions

Revision as of 23:34, 5 August 2016

Revision as of 22:16, 8 August 2016

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==Polynomials==

{{main article|Orthogonal polynomials}}

If one begins with the [[polynomialmononomial]] sequence {1, ''x'', ''x''², ... ''x''ⁿ ...} on [–1, 1] and applies the [[Gram-Schmidt process]], then one obtains the [[Legendre polynomial]]s. Another collection of orthogonal polynomials are the [[associated Legendre polynomials]].

The study of orthogonal polynomials involves a [[weight function]]s ''w''(''x'') which isare inserted in the bilinear form :

:<math> \langle f,g\rangle = \int w(x) f(x) g(x)\,dx .</math>

For [[Laguerre polynomial]]s on (0, ∞) the weight function is <math>w(x) = e^{-x} .</math>

Both probabilistsphysicists and physicistsprobability theorists use [[Hermite polynomial]]s on (−∞, ∞) where the weight function is <math>w(x) = e^{-x^2}</math> or <math>w(x) = e^{- \frac {x^2}{2}} .</math>

[[Chebyshev polynomial]]s are defined on [−1, 1] and use weights <math>w(x) = \frac{1}{\sqrt{1 - x^2}}</math> or <math>w(x) = \sqrt{1 - x^2}</math>