Truncated power function
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In mathematics, the truncated power function with exponent n {\displaystyle n} is defined as
x + n = { x n : x > 0 0 : x ≤ 0. {\displaystyle x_{+}^{n}={\begin{cases}x^{n}&:\ x>0\\0&:\ x\leq 0.\end{cases}}}
In particular,
x + = { x : x > 0 0 : x ≤ 0. {\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.
- x ↦ x + 0 {\displaystyle x\mapsto x_{+}^{0}} is the Heaviside function.
- χ [ a , b ) ( x ) = ( b − x ) + 0 − ( a − x ) + 0 {\displaystyle \chi _{[a,b)}(x)=(b-x)_{+}^{0}-(a-x)_{+}^{0}} where χ {\displaystyle \chi } is the indicator function.
- Truncated power functions are refinable.