Decomposition of spectrum (functional analysis)

Definition

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

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

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

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

and interpret the exponent as conventional power.

Relations

• Truncated power functions can be used for construction of B-Splines.
• $x\mapsto x_{+}^{0}$ 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