Triangle Function

$$\Lambda(x) \equiv \left\{\begin{array}{ll} 0 & \mbox{$\vert x\vert > 1$}\\ 1-\vert x\vert & \mbox{$\vert x\vert < 1$}\end{array}\right.$$ (1)

$$= \Pi(x)*\Pi(x)$$ (2)

$$= \Pi(x)*H(x+{\textstyle{1\over 2}})-\Pi(x)*H(x-{\textstyle{1\over 2}}),$$ (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:

$$\lambda(x,y,z)\equiv x^2+y^2+z^2-2xy-2xz-2yz.$$ (4)

It follows that

$$\lambda(a^2,b^2,c^2)=(a+b+c)(a+b-c)(a-b+c)(a-b-c).$$ (5)

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