Truncated power function

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

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

In particular,

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

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.

See also

• Macaulay brackets

External links

• Truncated Power Function on MathWorld

References

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