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where x j < x j + 1 {\displaystyle x_{j}<x_{j+1}} for all integer j {\displaystyle j} . The piecewise linear function passes through every point expressed as coordinates with ordered pair ( x j , y j ) {\displaystyle (x_{j},y_{j})} , that is,

f ( x j ) = y j {\displaystyle f(x_{j})=y_{j}} .

For any parameter a ≠ ≠ 0 {\displaystyle a\neq 0} :

tri ⁡ ⁡ ( t a ) = ( 1 a ) rect ⁡ ⁡ ( t a ) ∗ ∗ ( 1 a ) rect ⁡ ⁡ ( t a ) = ∫ ∫ − − ∞ ∞ ∞ ∞ 1 | a | rect ⁡ ⁡ ( τ τ a ) ⋅ ⋅ rect ⁡ ⁡ ( t − − τ τ a ) d τ τ = { 1 − − | t / a | , | t | < | a | ; 0 otherwise . {\displaystyle {\begin{aligned}\operatorname {tri} \left({\tfrac {t}{a}}\right)&=\left({\tfrac {1}{\sqrt {a}}}\right)\operatorname {rect} \left({\tfrac {t}{a}}\right)*\left({\tfrac {1}{\sqrt {a}}}\right)\operatorname {rect} \left({\tfrac {t}{a}}\right)=\int _{-\infty }^{\infty }{\tfrac {1}{|a|}}\operatorname {rect} \left({\tfrac {\tau }{a}}\right)\cdot \operatorname {rect} \left({\tfrac {t-\tau }{a}}\right)\,d\tau \\&={\begin{cases}1-|t/a|,&|t|<|a|;\\0&{\text{otherwise}}.\end{cases}}\end{aligned}}}

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