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:
:
The functions
and
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.
whenever
. 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
. It follows that the sequence
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
when
and ''n'' and ''m'' are positive integers. For then
:
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
on the interval
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
that are inserted in the bilinear form:
:
For
Laguerre polynomials on
the weight function is
.
Both physicists and probability theorists use
Hermite polynomials on
, where the weight function is
or
.
Chebyshev polynomials are defined on
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