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 ...

, a basis function is an element of a particular basis for 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 ...

. Every function 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 ''C^ω''

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 expon ...

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, 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 exampl ...

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 ...
)
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

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

Examples

Monomial basis for ''Cω''

Monomial basis for polynomials

See also

References

Monomial basis for ''C^ω''