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The rectangle function can be used to represent the Dirac delta function δ δ ( x ) {\displaystyle \delta (x)} .cite-ref-0-11-0[11] Specifically, δ δ ( x ) = lim a → → 0 1 a rect ⁡ ⁡ ( x a ) . {\displaystyle \delta (x)=\lim _{a\to 0}{\frac {1}{a}}\operatorname {rect} \left({\frac {x}{a}}\right).} For a function g ( x ) {\displaystyle g(x)} , its average over the width a {\displaystyle a} around 0 in the function domain is calculated as,

g a v g ( 0 ) = 1 a ∫ ∫ − − ∞ ∞ ∞ ∞ d x g ( x ) rect ⁡ ⁡ ( x a ) . {\displaystyle g_{avg}(0)={\frac {1}{a}}\int \limits _{-\infty }^{\infty }dx\ g(x)\operatorname {rect} \left({\frac {x}{a}}\right).} To obtain g ( 0 ) {\displaystyle g(0)} , the following limit is applied,

g ( 0 ) = lim a → → 0 1 a ∫ ∫ − − ∞ ∞ ∞ ∞ d x g ( x ) rect ⁡ ⁡ ( x a ) {\displaystyle g(0)=\lim _{a\to 0}{\frac {1}{a}}\int \limits _{-\infty }^{\infty }dx\ g(x)\operatorname {rect} \left({\frac {x}{a}}\right)} and this can be written in terms of the Dirac delta function as, g ( 0 ) = ∫ ∫ − − ∞ ∞ ∞ ∞ d x g ( x ) δ δ ( x ) . {\displaystyle g(0)=\int \limits _{-\infty }^{\infty }dx\ g(x)\delta (x).} The Fourier transform of the Dirac delta function δ δ ( t ) {\displaystyle \delta (t)} is

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