Truncated power function

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

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

In particular,

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

and interpret the exponent as conventional power.

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

• Macaulay brackets

• Truncated Power Function on MathWorld

• Numerical analysis