Truncated power function: Difference between revisions

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GivenIn a function ''f''mathematics, the '''truncated power function'''<ref>{{cite is defined asbook

{{references|date=October 2009}}

|title=Interpolation and Approximation with Splines and Fractals

{{expert-subject|Mathematics|date=October 2009}}

|first=Peter|last=Massopust

In the [[mathematical]] subfield of [[numerical analysis]] the '''truncated power function''' is a generalization of the [[indicator function]].

|publisher= Oxford University Press, USA

|year=2010

|isbn=978-0-19-533654-2

|page=46

}}</ref> with exponent <math>n</math> is defined as

:<math>f_x_+^n :=

==Definition==

\left\{\begin{matrixcases}

Given a function ''f'' the '''truncated power function''' is defined as

fx^n &\mbox{if}:\ fx \ge> 0 \\

0 &:\ x \le 0.

\end{cases}

</math>

In particular,

:<math>f_+^n :=

:<math>x_+ =

\left\{\begin{matrix}

\begin{cases}

f^n &\mbox{if}\ f \ge 0 \\

0x &\mbox{if}:\ fx <> 0 \\

0 &:\ x \le 0.

\end{matrix}\right.

\end{cases}

</math>

and interpret the exponent as conventional [[power function|power]].

==NotesRelations==

* Truncated power functions can be used for construction of [[B-spline]]s.

:<math>\chi_{(a,b]}(x) = (b-x)_+^0 - (a-x)_+^0</math>

* <math>x \mapsto x_+^0</math> is the [[Heaviside function]].

:* <math>\chi_{([a,b])}(x) = (b-x)_+^0 - (a-x)_+^0</math> where <math>\chi</math> is the [[indicator function]].

* Truncated power functions are [[refinable function|refinable]].

== See also ==

* [[Macaulay brackets]]

==External links==

*[http://mathworld.wolfram.com/TruncatedPowerFunction.html Truncated Power Function on MathWorld]

==References==

[[Category:Numerical analysis]]

<references/>

[[Category:Numerical analysis]]

{{math-stub}}