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:
:
The functions
and
are
orthogonal when this integral is zero, i.e.
whenever
. 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
. It follows that the sequence
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
when
and ''n'' and ''m'' are positive integers. For then
:
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
on the interval