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• Definitions

• Scaling

• Fourier transform

• See also

• References

The most common definition is as a piecewise function:

tri ⁡ ⁡ ( x ) = Λ Λ ( x ) = def max ( 1 − − | x | , 0 ) = { 1 − − | x | , | x | < 1 ; 0 otherwise . {\displaystyle {\begin{aligned}\operatorname {tri} (x)=\Lambda (x)\ &{\overset {\underset {\text{def}}{}}{=}}\ \max {\big (}1-|x|,0{\big )}\\&={\begin{cases}1-|x|,&|x|<1;\\0&{\text{otherwise}}.\\\end{cases}}\end{aligned}}}

Equivalently, it may be defined as the convolution of two identical unit rectangular functions:

tri ⁡ ⁡ ( x ) = rect ⁡ ⁡ ( x ) ∗ ∗ rect ⁡ ⁡ ( x ) = ∫ ∫ − − ∞ ∞ ∞ ∞ rect ⁡ ⁡ ( x − − τ τ ) ⋅ ⋅ rect ⁡ ⁡ ( τ τ ) d τ τ . {\displaystyle {\begin{aligned}\operatorname {tri} (x)&=\operatorname {rect} (x)*\operatorname {rect} (x)\\&=\int _{-\infty }^{\infty }\operatorname {rect} (x-\tau )\cdot \operatorname {rect} (\tau )\,d\tau .\\\end{aligned}}}

The triangular function can also be represented as the product of the rectangular and absolute value functions:

tri ⁡ ⁡ ( x ) = rect ⁡ ⁡ ( x / 2 ) ( 1 − − | x | ) . {\displaystyle \operatorname {tri} (x)=\operatorname {rect} (x/2){\big (}1-|x|{\big )}.}

Note that some authors instead define the triangle function to have a base of width 1 instead of width 2:

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