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Triangle Function


$\displaystyle \Lambda(x)$ $\textstyle \equiv$ $\displaystyle \left\{\begin{array}{ll} 0 & \mbox{$\vert x\vert > 1$}\\  1-\vert x\vert & \mbox{$\vert x\vert < 1$}\end{array}\right.$ (1)
  $\textstyle =$ $\displaystyle \Pi(x)*\Pi(x)$ (2)
  $\textstyle =$ $\displaystyle \Pi(x)*H(x+{\textstyle{1\over 2}})-\Pi(x)*H(x-{\textstyle{1\over 2}}),$ (3)

where $\Pi$ is the Rectangle Function and $H$ is the Heaviside Step Function. An obvious generalization used as an Apodization Function goes by the name of the Bartlett Function.

There is also a three-argument function known as the triangle function:

\lambda(x,y,z)\equiv x^2+y^2+z^2-2xy-2xz-2yz.
\end{displaymath} (4)

It follows that
\end{displaymath} (5)

See also Absolute Value, Bartlett Function, Heaviside Step Function, Ramp Function, Sgn, Triangle Coefficient

© 1996-9 Eric W. Weisstein