Orthogonal functions

Template:Short description 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> \langle f,g\rangle = \int \overline{f(x)}g(x)\,dx .</math>

The functions <math>f</math> and <math>g</math> are orthogonal when this integral is zero, i.e. <math>\langle f, \, g \rangle = 0</math> whenever <math>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> \{ f_0, f_1, \ldots\}</math> is a sequence of orthogonal functions of nonzero L²-norms <math display="inline"> \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>\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.

Trigonometric functions

Template:Main article Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions Template:Nowrap and Template:Nowrap are orthogonal on the interval <math>x \in (-\pi, \pi)</math> when <math>m \neq n</math> and n and m are positive integers. For then

<math>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.<ref>Antoni Zygmund (1935) Trigonometrical Series, page 6, Mathematical Seminar, University of Warsaw</ref> 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

Template:Main article If one begins with the monomial sequence <math> \left\{1, x, x^2, \dots\right\} </math> on the interval <math>[-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>w(x)</math> that are inserted in the bilinear form:

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

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

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

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

File:ChebychevRational1.png

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 Template:Nowrap while occasionally orthogonal families are required on Template:Nowrap. In this case it is convenient to apply the Cayley transform first, to bring the argument into Template:Nowrap. 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

• George B. Arfken & Hans J. Weber (2005) Mathematical Methods for Physicists, 6th edition, chapter 10: Sturm-Liouville Theory — Orthogonal Functions, Academic Press.
• Template:Cite journal
• Giovanni Sansone (translated by Ainsley H. Diamond) (1959) Orthogonal Functions, Interscience Publishers.

External links

• Orthogonal Functions, on MathWorld.