What are the elementary properties of dirac-delta function from which every other properties of it could be deduced?

Question

I am studying dirac-delta function first time in my undergraduate course and different books have defined this function in different ways which when graphed together contradicts each other. I want to know, if there is any, the elementary properties of this function from which every other properties of it could be deduced rigorously without "visualising" anything. I don't want to visualise this function because it does not make sense technically since it has infinite value at one point and zero elsewhere.

Answer 1

The only completely correct definition of Dirac delta is that it is the evaluation operator at some point ($0$ as default if not indicated otherwise). $$ \langle\delta, f\rangle=f(0). $$

This can be seen as a linear functional on the space of continuous functions or as a distribution on the space of test functions. Other sensible formulations that are equivalent is a point mass measure or a Riemann-Stieljes integral with a step function, $$ f(0)=\int_{\Bbb R}f(x)dH(x),~~~ H(x)=\begin{cases}0,& x<0,\\1,&x\ge0.\end{cases} $$

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There is a desire to work with the Dirac delta as a function, so that the above operator application becomes a scalar product. This is obviously not possible, as the integral operation is ignorant of values at single points.

What is possible is to approximate this behavior, this is called "approximation of unity" (in the sense of the convolution product), see for instance Carl Offner: "A little harmonic analysis" (Online PDF article). A sequence $\phi_n$ is such an approximation if

• all functions are non-negative, $\phi_n\ge 0$,
• the sequence converges point-wise to zero, $\phi_n(x)\to 0$ for $n\to\infty$ and $x\ne 0$, and
• the functions are integrable with integral one, $\int_{\Bbb R}\phi_n(x)\,dx=1$.

One can show that this gives convergence in the distributional sense to $\delta$. Note that other approximations in the distributional sense of $\delta$ exist. One could, for instance, give examples with $\phi_n(0)=0$.