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In mathematics, particularly numerical analysis, a basis function is an element of the basis for a function space. The use of the term is analogous to basis vector for a vector space.

To be expanded later A basis set is a collection of fundamental functions in terms of which another set of functions can be expressed. For example, in a Fourier series expansion, the constituent sinusoids make up the basis set in terms of which arbitrary waveforms can be expressed. The most useful type of basis set for practical problems—particularly in the problem space of signal reception and estimation and optimum filtration—is the orthonormal basis set, which can be derived from any arbitrary set of functions via the procedure known as Gram-Schmidt orthogonalization.

[edit] Example

The collection of quadratic polynomials with real coefficients has {1,t,t²} as a basis. Every quadratic can be written as a1+bt+ct², that is, as a linear combination of the basis functions 1, t, and t². The set {(1/2)(t-1)(t-2),-t(t-2),(1/2)t(t-1)} is another basis for quadratic polynomials called the Lagrange basis).

[edit] Old article

In functional analysis and its applications, a function space can be viewed as a vector space of infinite dimension whose basis vectors are functions. This means that each function in the function space can be represented as a linear combination of the basis functions.

Let's illustrate the concept of basis using a simple example. You can create any (two-dimensional) vector, (x,y) by adding multiples of the vectors (1,0) and (0,1):

<math>

\begin{bmatrix}x\\y\end{bmatrix} = x \begin{bmatrix}1\\0\end{bmatrix} + y\begin{bmatrix}0\\1\end{bmatrix}. </math> In this example, we say that the vector (x,y) is in the space spanned by the vectors (1,0) and (0,1). The most convenient basis vectors are perpendicular or orthogonal to each other, which is true of (1,0) and (0,1). Two vectors are orthogonal if their scalar product is zero, which means they are at right angles. Likewise, two functions are orthogonal if their inner product is zero. Sine and cosine are orthogonal functions because

<math>\int_{-\infty}^{+\infty}\sin(x)\cos(x)dx=0</math>.

A function f(x) is square integrable if and only if

<math>\int_{-\infty}^{+\infty}|f(x)|^2 dx < \infty</math>.

Any square integrable function (for example a musical recording) can be represented by a sum of sines and cosines of various amplitudes and frequencies. This is termed the function's corresponding Fourier series. If the function requires a continuous range of frequencies for its representation—a discrete set being insufficient for what is not ultimately a periodic function—then the frequency-domain representation is termed the Fourier transform. In this example, the sines and cosines are the basis functions. Note that while the two-dimensional plane is spanned by only two basis vectors, a function space is spanned by an infinite number of basis functions, because the function space is infinite-dimensional.es:Función base pt:Função de base