Truncated power function

In mathematics, the truncated power function¹ 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.

• 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.