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Recall that a $\delta$-sequence can be defined as a sequence, $(\phi_n)_{n\in\mathbb N}$, of continuously differentiable, non-negative, real-valued functions for which $\int_\mathbb R\phi_ndx=1$ for all $n\in\mathbb N$. An example of a $\delta$-sequence often discussed on this website is that of, $$\phi_n(x)=\frac{n}{\sqrt\pi}e^{-n^2x^2},$$ where $n\in\mathbb N$ and $x\in\mathbb R$. Since $\phi_n(x)\to0$ as $n\to\infty$ for $|x|\to\infty$, we see that $(\phi_n)_{n\in\mathbb N}$ is a sequence of functions which vanish at infinity.

What is an example of a $\delta$-sequence which satisfies the stronger requirement of having compact support for each $n\in\mathbb N$?

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Let $\phi_n(x)=\frac 1 n f(\frac x n)$ where $f(x)=c(x+1)^{2}(1-x)^{2}$ for $-1\leq x \leq 1$, with $f(x)=0$ for $x \notin [-1,1]$. Choose the positive constant $c$ such that $\int f(x)\, dx=1$.

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  • $\begingroup$ For this to be a $\delta$-sequence in the usual sense, shouldn't you take $\phi_n(x)=nf(nx)$? Your sequence satisfies the requirements of the opening post, though. $\endgroup$ – Mars Plastic Jan 31 '19 at 13:09
  • $\begingroup$ What do you mean "in the usual sense"? I concede that my definition about is somewhat rough. $\endgroup$ – Jeremy Jeffrey James Jan 31 '19 at 13:40
  • $\begingroup$ For any continuous function $g\colon\mathbb R\to\mathbb R$, the term $\int_{\mathbb R} \phi_n(x)g(x) d(x)$ should converge to $g(0)$. In particular, if you think of $\phi_n$ as a regular distribution, it should converge strongly to the $\delta$-distribution in 0. This is the main reason why it's called a $\delta$-sequence. $\endgroup$ – Mars Plastic Jan 31 '19 at 16:29
  • $\begingroup$ @MarsPlastic $\delta-$ sequence is not a standard terminology. Could you state all the properties you want from the sequence precisely? $\endgroup$ – Kavi Rama Murthy Jan 31 '19 at 23:14
  • $\begingroup$ I don't think there is a general consensus about a precise definition (I may be wrong, though), but the term "$\delta$-sequence" itself suggests that it should at the very least converge to a $\delta$-distribution (pointwise when interpreted as a sequence of regular distributions). However, as I said: Concerning the requirements as stated in the opening post, your answer is undoubtedly correct. $\endgroup$ – Mars Plastic Feb 1 '19 at 12:20

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