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Triangle Function

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TriangleFunction

The triangle function is the function

Lambda(x)={0 |x|>=1; 1-|x| |x|<1
(1)
=Pi(x)*Pi(x)
(2)
=Pi(x)*H(x+1/2)-Pi(x)*H(x-1/2),
(3)

where Pi(x) is the rectangle function, H(x) is the Heaviside step function, and * denotes convolution. An obvious generalization used as an apodization function goes by the name of the Bartlett function.

The piecewise version of the triangle function is implemented in the Wolfram Language as UnitTriangle[x], while the generalized function version is implemented as HeavisideLambda[x].

TriangleFunction3D

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

lambda(x,y,z)=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, Rectangle Function, Sign, Triangle Coefficient, Triangle Wave, Triangular Distribution

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References

Bracewell, R. "The Triangle Function of Unit Height and Area, Lambda(x)." In The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, p. 53, 1999.

Referenced on Wolfram|Alpha

Triangle Function

Cite this as:

Weisstein, Eric W. "Triangle Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TriangleFunction.html

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