Orthogonal functions: Difference between revisions

Revision as of 01:33, 8 March 2018

Revision as of 01:37, 8 March 2018

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==Polynomials==

{{main article|Orthogonal polynomials}}

If one begins with the [[monomial]] sequence {1, ''x'', ''x''², ..., ''x''^''n'', ...} on {{nowrap|[–1, 1]}} and applies the [[Gram-SchmidtGram–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 [[weight function]]s ''w''(''x'') whichthat are inserted in the bilinear form:

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

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

Both physicists and probability theorists use [[Hermite polynomial]]s on {{nowrap|(−∞, ∞)}}, 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 {{nowrap|[−1, 1]}} and use weights <math>w(x) = \frac{1}{\sqrt{1 - x^2}}</math> or <math>w(x) = \sqrt{1 - x^2}</math>.

[[Zernike polynomial]]s are defined on the [[unit disk]] and have orthogonality of both radial and angular parts.