Truncated power function: Difference between revisions

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In the [[mathematical]] subfield of [[numerical analysis]]mathematics, the '''truncated power function'''<ref>{{cite is a generalization of the [[indicator function]].book

|title=Interpolation and Approximation with Splines and Fractals

|first=Peter|last=Massopust

|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}

x^n &:\ x > 0 \\

0 &:\ x \le 0.

\end{cases}

</math>

In particular,

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

:<math>x_+ =

\begin{cases}

:<math>f_+^n :=

x &:\ x > 0 \\

\left\{\begin{matrix}

f^n0 &\mbox{if}:\ fx \gele 0 \\.

\end{cases}

0 &\mbox{if}\ f < 0

\end{matrix}\right.

</math>

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

==Relations==

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

* <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]].

==Notes See also ==

* [[Macaulay brackets]]

==External links==

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

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

==External LinksReferences==

<references/>

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

[[Category:Numerical analysis]]