Orthogonal functions

Short description: Type of function

In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval:

[math]\displaystyle{ \langle f,g\rangle = \int \overline{f(x)}g(x)\,dx . }[/math]

The functions [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] are orthogonal when this integral is zero, i.e. [math]\displaystyle{ \langle f, \, g \rangle = 0 }[/math] whenever [math]\displaystyle{ f \neq g }[/math]. As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the above integral is the equivalent of a vector dot product; two vectors are mutually independent (orthogonal) if their dot-product is zero.

Suppose [math]\displaystyle{ \\{ f_0, f_1, \ldots\\} }[/math] is a sequence of orthogonal functions of nonzero L2-norms [math]\displaystyle{ \left\| f_n \right\| _2 = \sqrt{\langle f_n, f_n \rangle} = \left(\int f_n ^2 \ dx \right) ^\frac{1}{2} }[/math]. It follows that the sequence [math]\displaystyle{ \left\\{ f_n / \left\| f_n \right\| _2 \right\\} }[/math] is of functions of L2-norm one, forming an orthonormal sequence. To have a defined L2-norm, the integral must be bounded, which restricts the functions to being square-integrable.

## Contents

* 1 Trigonometric functions
* 2 Polynomials
* 3 Binary-valued functions
* 4 Rational functions
* 5 In differential equations
* 6 See also
* 7 References
* 8 External links

## Trigonometric functions

Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions sin nx and sin mx are orthogonal on the interval [math]\displaystyle{ x \in (-\pi, \pi) }[/math] when [math]\displaystyle{ m \neq n }[/math] and n and m are positive integers. For then

[math]\displaystyle{ 2 \sin \left(mx\right) \sin \left(nx\right) = \cos \left(\left(m - n\right)x\right) - \cos\left(\left(m+n\right) x\right), }[/math]

and the integral of the product of the two sine functions vanishes.[1] Together with cosine functions, these orthogonal functions may be assembled into a trigonometric polynomial to approximate a given function on the interval with its Fourier series.

## Polynomials

If one begins with the monomial sequence [math]\displaystyle{ \left\\{1, x, x^2, \dots\right\\} }[/math] on the interval [math]\displaystyle{ [-1,1] }[/math] and applies the Gram–Schmidt process, then one obtains the Legendre polynomials. Another collection of orthogonal polynomials are the associated Legendre polynomials.

The study of orthogonal polynomials involves weight functions [math]\displaystyle{ w(x) }[/math] that are inserted in the bilinear form:

[math]\displaystyle{ \langle f,g\rangle = \int w(x) f(x) g(x)\,dx . }[/math]

For Laguerre polynomials on [math]\displaystyle{ (0,\infty) }[/math] the weight function is [math]\displaystyle{ w(x) = e^{-x} }[/math].

Both physicists and probability theorists use Hermite polynomials on [math]\displaystyle{ (-\infty,\infty) }[/math], where the weight function is [math]\displaystyle{ w(x) = e^{-x^2} }[/math] or [math]\displaystyle{ w(x) = e^{- x^2/2} }[/math].

Chebyshev polynomials are defined on [math]\displaystyle{ [-1,1] }[/math] and use weights [math]\displaystyle{ w(x) = \frac{1}{\sqrt{1 - x^2}} }[/math] or [math]\displaystyle{ w(x) = \sqrt{1 - x^2} }[/math].

Zernike polynomials are defined on the unit disk and have orthogonality of both radial and angular parts.

## Binary-valued functions

Walsh functions and Haar wavelets are examples of orthogonal functions with discrete ranges.

## Rational functions

Plot of the Chebyshev rational functions of order n=0,1,2,3 and 4 between x=0.01 and 100.

Legendre and Chebyshev polynomials provide orthogonal families for the interval [−1, 1] while occasionally orthogonal families are required on [0, ∞). In this case it is convenient to apply the Cayley transform first, to bring the argument into [−1, 1]. This procedure results in families of rational orthogonal functions called Legendre rational functions and Chebyshev rational functions.

## In differential equations

Solutions of linear differential equations with boundary conditions can often be written as a weighted sum of orthogonal solution functions (a.k.a. eigenfunctions), leading to generalized Fourier series.

## See also

* Eigenvalues and eigenvectors
* Hilbert space
* Karhunen–Loève theorem
* Lauricella's theorem
* Wannier function

## References

1. ↑ Antoni Zygmund (1935) Trigonometrical Series, page 6, Mathematical Seminar, University of Warsaw

* George B. Arfken & Hans J. Weber (2005) Mathematical Methods for Physicists, 6th edition, chapter 10: Sturm-Liouville Theory — Orthogonal Functions, Academic Press.
* Price, Justin J. (1975). "Topics in orthogonal functions". American Mathematical Monthly 82: 594–609. doi:10.2307/2319690. http://www.maa.org/programs/maa-awards/writing-awards/topics-in-orthogonal-functions.
* Giovanni Sansone (translated by Ainsley H. Diamond) (1959) Orthogonal Functions, Interscience Publishers.

## External links

* Orthogonal Functions, on MathWorld.

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