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.

External links

Truncated Power Function on MathWorld

References

↑ Massopust, Peter (2010). Interpolation and Approximation with Splines and Fractals. Oxford University Press, USA. p. 46. ISBN 0-19-533654-2.