Approximating a Delta Function with Isoceles Triangles
The delta function is defined so that
\[
\delta(x-a)=
\begin{cases}
0, &x\ne a\\
\infty, & x=a
\end{cases}
\]
and normalized so that
\[\int_{-\infty}^{\infty} \delta(x-a)\, dx =1\].
Find a set of functions that approximate the delta function \(\delta(x-a)\) with a sequence of isosceles triangles \(\delta_{\epsilon}(x-a)\), centered at \(a\), that get narrower and taller as the parameter \(\epsilon\) approaches zero.
Choose the approximation functions so that their integral
over all space is always equal to one.
Using the test function \(f(x)=3x^2\), find the value of
\[\int_{-\infty}^{\infty} f(x)\delta_{\epsilon}(x-a)\, dx\]
Then, show that the integral approaches \(f(a)\) in the limit that \(\epsilon \rightarrow 0\).