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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, orthogonal functions belong to a
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
that is a vector space equipped with a
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
. When the function space has an interval as the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
, the bilinear form may be the integral of the product of functions over the interval: : \langle f,g\rangle = \int \overlineg(x)\,dx . The functions f and g are
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
when this integral is zero, i.e. \langle f, \, g \rangle = 0 whenever f \neq g. 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 \ is a sequence of orthogonal functions of nonzero ''L''2-norms \left\, f_n \right\, _2 = \sqrt = \left(\int f_n ^2 \ dx \right) ^\frac . It follows that the sequence \left\ is of functions of ''L''2-norm one, forming an
orthonormal sequence In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
. To have a defined ''L''2-norm, the integral must be bounded, which restricts the functions to being square-integrable.


Trigonometric functions

Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions and are orthogonal on the interval x \in (-\pi, \pi) when m \neq n and ''n'' and ''m'' are positive integers. For then :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), and the integral of the product of the two sine functions vanishes. Antoni Zygmund (1935) ''
Trigonometrical Series In mathematics, a trigonometric series is a infinite series of the form : \frac+\displaystyle\sum_^(A_ \cos + B_ \sin), an infinite version of a trigonometric polynomial. It is called the Fourier series of the integrable function f if the ter ...
'', page 6, Mathematical Seminar, University of Warsaw
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 A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
.


Polynomials

If one begins with the monomial sequence \left\ on the interval 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 In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation \left(1 - x^2\right) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0, or equivalently ...
. The study of orthogonal polynomials involves weight functions w(x) that are inserted in the bilinear form: : \langle f,g\rangle = \int w(x) f(x) g(x)\,dx . For Laguerre polynomials on (0,\infty) the weight function is w(x) = e^. Both physicists and probability theorists use Hermite polynomials on (-\infty,\infty), where the weight function is w(x) = e^ or w(x) = e^. Chebyshev polynomials are defined on 1,1/math> and use weights w(x) = \frac or w(x) = \sqrt. Zernike polynomials are defined on the
unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose di ...
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

Legendre and Chebyshev polynomials provide orthogonal families for the interval while occasionally orthogonal families are required on . In this case it is convenient to apply the Cayley transform first, to bring the argument into . 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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
* 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. * {{cite journal, author=Price, Justin J., authorlink=Justin Jesse Price, title=Topics in orthogonal functions, journal=
American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an e ...
, volume=82, year=1975, pages=594–609, url=http://www.maa.org/programs/maa-awards/writing-awards/topics-in-orthogonal-functions, doi=10.2307/2319690 *
Giovanni Sansone Giovanni Sansone (24 May 1888 – 13 October 1979) was an Italian mathematician, known for his works on mathematical analysis, on the theory of orthogonal functions and on the theory of ordinary differential equations. He was an Invited Speaker ...
(translated by Ainsley H. Diamond) (1959) ''Orthogonal Functions'', Interscience Publishers.


External links


Orthogonal Functions
on MathWorld. Functional analysis Types of functions