Truncated power function

In mathematics, the truncated power function^[1] with exponent ${\displaystyle n}$ is defined as

${\displaystyle x_{+}^{n}={\begin{cases}x^{n}&:\ x>0\\0&:\ x\leq 0.\end{cases}}}$

In particular,

${\displaystyle x_{+}={\begin{cases}x&:\ x>0\\0&:\ x\leq 0.\end{cases}}}$

and interpret the exponent as conventional power.

Relations

• Truncated power functions can be used for construction of B-splines.
• ${\displaystyle x\mapsto x_{+}^{0}}$  is the Heaviside function.
• ${\displaystyle \chi _{[a,b)}(x)=(b-x)_{+}^{0}-(a-x)_{+}^{0}}$  where ${\displaystyle \chi }$  is the indicator function.
• Truncated power functions are refinable.

See also

• Macaulay brackets

External links

• Truncated Power Function on MathWorld

References

1. Massopust, Peter (2010). Interpolation and Approximation with Splines and Fractals. Oxford University Press, USA. p. 46. ISBN 0-19-533654-2.