Dirac delta function
Approximations to the identity

Typically a nascent delta function η can be constructed in the following manner. Let η be an absolutely integrable function on R of total integral 1, and define η ε (x) = ε − 1 η (x ε). {\displaystyle \eta _{\varepsilon }(x)=\varepsilon ^{-1}\eta \left({\frac {x}{\varepsilon }}\right).}

In n dimensions, one uses instead the scaling η ε (x) = ε − n η (x ε). {\displaystyle \eta _{\varepsilon }(x)=\varepsilon ^{-n}\eta \left({\frac {x}{\varepsilon }}\right).}

Then a simple change of variables shows that η also has integral 1. One may show that (5) holds for all continuous compactly supported functions f, and so η converges weakly to δ in the sense of measures.

The η constructed in this way are known as an approximation to the identity. This terminology is because the space L (R) of absolutely integrable functions is closed under the operation of convolution of functions: f ∗ g ∈ L (R) whenever f and g are in L (R). However, there is no identity in L (R) for the convolution product: no element h such that f ∗ h = f for all f. Nevertheless, the sequence η does approximate such an identity in the sense that

f ∗ η ε → f as ε → 0. {\displaystyle f*\eta _{\varepsilon }\to f\quad {\text{as }}\varepsilon \to 0.}

This limit holds in the sense of mean convergence (convergence in L). Further conditions on the η, for instance that it be a mollifier associated to a compactly supported function, are needed to ensure pointwise convergence almost everywhere.

If the initial η = η is itself smooth and compactly supported then the sequence is called a mollifier. The standard mollifier is obtained by choosing η to be a suitably normalized bump function, for instance

η (x) = { e − 1 1 − | x | 2 if | x | < 1 0 if | x | ≥ 1. {\displaystyle \eta (x)={\begin{cases}e^{-{\frac {1}{1-|x|^{2}}}}&{\text{if }}|x|<1\\0&{\text{if }}|x|\geq 1.\end{cases}}}

In some situations such as numerical analysis, a piecewise linear approximation to the identity is desirable. This can be obtained by taking η to be a hat function. With this choice of η, one has

η ε (x) = ε − 1 max (1 − | x ε |, 0) {\displaystyle \eta _{\varepsilon }(x)=\varepsilon ^{-1}\max \left(1-\left|{\frac {x}{\varepsilon }}\right|,0\right)}

which are all continuous and compactly supported, although not smooth and so not a mollifier.
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