Decomposition of spectrum (functional analysis)

Definition

The truncated power function with exponent $n$ is defined as

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

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

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

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