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In mathematics, orthogonal functions belong to a function space that is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
of the product of functions over the interval: : \langle f,g\rangle = \int \overlineg(x)\,dx . The functions f and g are orthogonal when this integral is zero, i.e. \langle f, \, g \rangle = 0 whenever f \neq g. As with a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
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. To have a defined ''L''2-norm, the integral must be bounded, which restricts the functions to being
square-integrable In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
.


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 Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century. ...
(1935) '' Trigonometrical Series'', page 6, Mathematical Seminar, University of Warsaw
Together with cosine functions, these orthogonal functions may be assembled into a
trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...
to approximate a given function on the interval with its Fourier series.


Polynomials

If one begins with the
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
sequence \left\ on the interval 1,1/math> and applies the
Gram–Schmidt process In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space equipped with the standard inner produ ...
, then one obtains the
Legendre polynomial In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
s. Another collection of orthogonal polynomials are the associated Legendre polynomials. The study of orthogonal polynomials involves
weight function A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is ...
s w(x) that are inserted in the bilinear form: : \langle f,g\rangle = \int w(x) f(x) g(x)\,dx . For
Laguerre polynomial In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions only ...
s on (0,\infty) the weight function is w(x) = e^. Both physicists and probability theorists use
Hermite polynomial In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as ...
s on (-\infty,\infty), where the weight function is w(x) = e^ or w(x) = e^.
Chebyshev polynomial The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...
s are defined on 1,1/math> and use weights w(x) = \frac or w(x) = \sqrt.
Zernike polynomial In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, winner of the 1953 Nobel Prize in Physics and the inventor of phase-contrast microscopy, th ...
s are defined on the unit disk and have orthogonality of both radial and angular parts.


Binary-valued functions

Walsh function In mathematics, more specifically in harmonic analysis, Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be used to represent any continuous ...
s and
Haar wavelet In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be repres ...
s 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 In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform ...
first, to bring the argument into . This procedure results in families of
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
orthogonal functions called
Legendre rational functions In mathematics the Legendre rational functions are a sequence of orthogonal functions on  , ∞). They are obtained by composing the Cayley transform with Legendre polynomials">Cayley_transform.html" ;"title=", ∞). They are obta ...
and
Chebyshev rational functions In mathematics, the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal. They are named after Pafnuty Chebyshev. A rational Chebyshev function of degree is defined as: :R_n(x)\ \stackrel\ T_n\left( ...
.


In differential equations

Solutions of linear
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s with boundary conditions can often be written as a weighted sum of orthogonal solution functions (a.k.a.
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s), leading to
generalized Fourier series In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space. Here we consider that of square-integrable functions de ...
.


See also

*
Eigenvalues and eigenvectors In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
* Hilbert space * Karhunen–Loève theorem *
Lauricella's theorem In the theory of orthogonal functions, Lauricella's theorem provides a condition for checking the closure of a set of orthogonal functions, namely: ''Theorem.'' A necessary and sufficient condition that a normal orthogonal set \ be closed is that ...
*
Wannier function The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier in 1937. Wannier functions are the localized molecular orbitals of crystalline systems. The Wannier functions fo ...


References

* George B. Arfken & Hans J. Weber (2005) ''Mathematical Methods for Physicists'', 6th edition, chapter 10: Sturm-Liouville Theory — Orthogonal Functions,
Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes referen ...
. * {{cite journal, author=Price, Justin J., authorlink=Justin Jesse Price, title=Topics in orthogonal functions, journal= American Mathematical Monthly, 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 (translated by Ainsley H. Diamond) (1959) ''Orthogonal Functions'',
Interscience Publishers John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in p ...
.


External links


Orthogonal Functions
on MathWorld. Functional analysis Types of functions