Truncated power function: Difference between revisions

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TheIn mathematics, the '''truncated power function'''<ref>{{cite book

==Definition==

The '''truncated power function'''<ref>{{cite book

|title=Interpolation and Approximation with Splines and Fractals

|first=Peter|last=Massopust

|publisher:= Oxford University Press, USA

|year=2010

|isbn=0195336542978-0-19-533654-2

|page=46

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

Line 11 ⟶ 10:

:<math>x_+^n =

\begin{cases}

x^n &:\ x \ge> 0 \\

0 &:\ x <\le 0.

\end{cases}

</math>

In particular,

Alternatively, you may consider the subscript plus as an individual function with

:<math>x_+ =

\begin{cases}

x &:\ x \ge> 0 \\

0 &:\ x <\le 0.

\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==