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| | I like my hobby Archery. Seems boring? Not at all!<br>I to learn German in my free time.<br><br>Also visit my webpage - [http://bme.med.tsinghua.edu.cn/wiki/index.php?title=Finding_Are_Employed_In_The_Task_Market_Today bme.med.tsinghua.edu.cn] |
| In [[mathematics]], a '''basis function''' is an element of a particular [[Basis (linear algebra)|basis]] for a [[function space]]. Every continuous function in the function space can be represented as a [[linear combination]] of basis functions, just as every vector in a [[vector space]] can be represented as a linear combination of [[basis vectors]].
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| In [[numerical analysis]] 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).
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| ==Examples==
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| ===Polynomial bases===
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| The collection of quadratic polynomials with real coefficients has {1, ''t'', ''t''<sup>2</sup>} as a basis. Every quadratic polynomial can be written as ''a''1+''bt''+''ct''<sup>2</sup>, that is, as a [[linear combination]] of the basis functions 1, ''t'', and ''t''<sup>2</sup>. 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 polynomial|Lagrange basis]].
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| ===Fourier basis===
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| Sines and cosines form an ([[orthonormality|orthonormal]]) [[Schauder basis]] for square-integrable functions. As a particular example, the collection:
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| :<math>\{\sqrt{2}\sin(2\pi n x) \; | \; n\in\mathbb{N} \} \cup \{\sqrt{2} \cos(2\pi n x) \; | \; n\in\mathbb{N} \} \cup\{1\}</math>
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| forms a basis for [[Lp space|L<sup>2</sup>(0,1)]].
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| ==References==
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| *{{cite book |last=Ito |first=Kiyoshi |authorlink= |coauthors= |others= |title=Encyclopedic Dictionary of Mathematics |edition=2nd ed. |year=1993 |publisher=MIT Press |location= |isbn=0-262-59020-4 | page=1141}}
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| ==See also==
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| {{col-begin}}
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| {{col-1-of-3}}
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| * [[Basis (linear algebra)]] ([[Hamel basis]])
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| * [[Schauder basis]] (in a [[Banach space]])
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| * [[Dual basis]]
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| * [[Biorthogonal system]] (Markushevich basis)
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| {{col-2-of-3}}
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| * [[Orthonormal basis]] in an [[inner-product space]]
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| * [[Orthogonal polynomials]]
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| * [[Fourier analysis]] and [[Fourier series]]
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| * [[Harmonic analysis]]
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| * [[Orthogonal wavelet]]
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| * [[Biorthogonal wavelet]]
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| {{col-3-of-3}}
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| * [[Radial basis function]] <!-- shape functions in the [[Galerkin method]] and -->
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| * [[Finite element analysis#Choosing a basis|Finite-elements (bases)]]
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| * [[Functional analysis]]
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| * [[Approximation theory]]
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| * [[Numerical analysis]]
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| {{col-end}}
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| [[Category:Numerical analysis]]
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| [[Category:Fourier analysis]]
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| [[Category:Linear algebra]]
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| [[Category:Numerical linear algebra]]
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| [[Category:Types of functions]]
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I like my hobby Archery. Seems boring? Not at all!
I to learn German in my free time.
Also visit my webpage - bme.med.tsinghua.edu.cn