# Truncated power function

## Definition

The truncated power function[1] with exponent $n$ is defined as

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

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

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

and interpret the exponent as conventional power.

## Relations

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