Editing Dirac delta function (section)

====Approximations to the identity====
Typically a nascent delta function {{mvar|η<sub>ε</sub>}} can be constructed in the following manner.  Let {{mvar|η}} be an absolutely integrable function on {{math|'''R'''}} of total integral {{math|1}}, and define
<math display="block">\eta_\varepsilon(x) = \varepsilon^{-1} \eta \left (\frac{x}{\varepsilon} \right). </math>

In {{mvar|n}} dimensions, one uses instead the scaling
<math display="block">\eta_\varepsilon(x) = \varepsilon^{-n} \eta \left (\frac{x}{\varepsilon} \right). </math>

Then a simple change of variables shows that {{mvar|η<sub>ε</sub>}} also has integral {{math|1}}.  One may show that ({{EquationNote|5}}) holds for all continuous compactly supported functions {{mvar|f}},{{sfn|Stein|Weiss|1971|loc=Theorem 1.18}} and so {{mvar|η<sub>ε</sub>}} converges weakly to {{mvar|δ}} in the sense of measures.

The {{mvar|η<sub>ε</sub>}} constructed in this way are known as an '''approximation to the identity'''.{{sfn|Rudin|1991|loc=§II.6.31}} This terminology is because the space {{math|''L''<sup>1</sup>('''R''')}} of absolutely integrable functions is closed under the operation of [[convolution]] of functions: {{math|''f'' ∗ ''g'' ∈ ''L''<sup>1</sup>('''R''')}} whenever {{mvar|f}} and {{mvar|g}} are in {{math|''L''<sup>1</sup>('''R''')}}.  However, there is no identity in {{math|''L''<sup>1</sup>('''R''')}} for the convolution product: no element {{mvar|h}} such that {{math|1=''f'' ∗ ''h'' = ''f''}} for all {{mvar|f}}.  Nevertheless, the sequence {{mvar|η<sub>ε</sub>}} does approximate such an identity in the sense that

<math display="block">f*\eta_\varepsilon \to f \quad \text{as }\varepsilon\to 0.</math>

This limit holds in the sense of [[mean convergence]] (convergence in {{math|''L''<sup>1</sup>}}).  Further conditions on the {{mvar|η<sub>ε</sub>}}, for instance that it be a mollifier associated to a compactly supported function,<ref>More generally, one only needs {{math|1=''η'' = ''η''<sub>1</sub>}} to have an integrable radially symmetric decreasing rearrangement.</ref> are needed to ensure pointwise convergence [[almost everywhere]].

If the initial {{math|1=''η'' = ''η''<sub>1</sub>}} is itself smooth and compactly supported then the sequence is called a [[mollifier]].  The standard mollifier is obtained by choosing {{mvar|η}} to be a suitably normalized [[bump function]], for instance

<math display="block">\eta(x) = \begin{cases}
\frac{1}{I_n} \exp\Big( -\frac{1}{1-|x|^2} \Big) & \text{if } |x| < 1\\
0                     & \text{if } |x|\geq 1.
\end{cases}</math>
(<math>I_n</math> ensuring that the total integral is 1).

In some situations such as [[numerical analysis]], a [[piecewise linear function|piecewise linear]] approximation to the identity is desirable.  This can be obtained by taking {{math|''η''<sub>1</sub>}} to be a [[hat function]].  With this choice of {{math|''η''<sub>1</sub>}}, one has

<math display="block"> \eta_\varepsilon(x) = \varepsilon^{-1}\max \left (1-\left|\frac{x}{\varepsilon}\right|,0 \right) </math>

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