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In mathematics, a basis function is an element of a particular
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 ...
for a function space. Every
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
in the function space can be represented as a linear combination of basis functions, just as every vector in 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 ...
can be represented as a linear combination of
basis vectors In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as componen ...
. In
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
and approximation theory, basis functions are also called blending functions, because of their use in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points).


Examples


Monomial basis for ''Cω''

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 ...
basis for the vector space of
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s is given by \. This basis is used in
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, amongst others.


Monomial basis for polynomials

The monomial basis also forms a basis for the vector space of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s. After all, every polynomial can be written as a_0 + a_1x^1 + a_2x^2 + \cdots + a_n x^n for some n \in \mathbb, which is a linear combination of monomials.


Fourier basis for ''L''2 ,1/h2>

Sines and cosines form an (
orthonormal 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 ...
)
Schauder basis In mathematics, a Schauder basis or countable basis is similar to the usual ( Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This ...
for square-integrable functions on a bounded domain. As a particular example, the collection \ \cup \ \cup \ forms a basis for ''L''2 ,1


See also

*
Basis (linear algebra) In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as component ...
(
Hamel basis In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
) *
Schauder basis In mathematics, a Schauder basis or countable basis is similar to the usual ( Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This ...
(in a Banach space) *
Dual basis In linear algebra, given a vector space ''V'' with a basis ''B'' of vectors indexed by an index set ''I'' (the cardinality of ''I'' is the dimension of ''V''), the dual set of ''B'' is a set ''B''∗ of vectors in the dual space ''V''∗ with the ...
*
Biorthogonal system In mathematics, a biorthogonal system is a pair of indexed families of vectors \tilde v_i \text E \text \tilde u_i \text F such that \left\langle\tilde v_i , \tilde u_j\right\rangle = \delta_, where E and F form a pair of topological vector spaces t ...
(Markushevich basis) *
Orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
in an
inner-product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
*
Orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the class ...
* Fourier analysis and Fourier series * Harmonic analysis *
Orthogonal wavelet An orthogonal wavelet is a wavelet whose associated wavelet transform is orthogonal. That is, the inverse wavelet transform is the adjoint of the wavelet transform. If this condition is weakened one may end up with biorthogonal wavelets. Basics ...
*
Biorthogonal wavelet A Biorthogonal wavelet is a wavelet where the associated wavelet transform is invertible but not necessarily orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogon ...
*
Radial basis function A radial basis function (RBF) is a real-valued function \varphi whose value depends only on the distance between the input and some fixed point, either the origin, so that \varphi(\mathbf) = \hat\varphi(\left\, \mathbf\right\, ), or some other fixed ...
* Finite-elements (bases) *
Functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
* Approximation theory *
Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...


References

*{{cite book , last=Itô , first=Kiyosi , title=Encyclopedic Dictionary of Mathematics , edition=2nd , year=1993 , publisher=MIT Press , isbn=0-262-59020-4 , page=1141 Numerical analysis Fourier analysis Linear algebra Numerical linear algebra Types of functions