Truncated power function

Definition

The truncated power function^[1] with exponent ${\displaystyle n}$  is defined as

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

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

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

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.

External links

• Truncated Power Function on MathWorld

References

1. Massopust, Peter (2010). Interpolation and Approximation with Splines and Fractals. p. 46. ISBN 0195336542. {{cite book}}: Text "publisher: Oxford University Press, USA" ignored (help)