Delta-Convergent Sequences

 
[Generalized Functions 01] I. M. Gel'fand p34 2.5. Delta-Convergent Sequences There are many ways to construct a sequence of regular functions which converge to the $\delta$ function. All that is needed is that the corresponding ordinary functions $f_\nu(x)$ form what we shall call a deltaconvergent sequence, which means that they must possess the following two properties. (a) For any $M>0$ and for $|a| \leqslant M$ and $|b| \leqslant M$, the quantities \[ \left|\int_a^b f_p(\xi) d \xi\right| \] must be bounded by a constant independent of $a, b$, or $v$ (in other words, depending only on $M$ ). (b) For any fixed nonvanishing $a$ and $b$, we must have \[ \lim _{v \rightarrow \infty} \int_a^b f_v(\xi) d \xi= \begin{cases}0 & \text { for } aIntegral and Series Representations of the Dirac Delta Function