0

How to prove that the series $$\sum_{k=-\infty}^{\infty}a_k\delta^{(k)}(x-k)$$ converges in $D'$ for all values of $a_k$?
I Understand that the partial sum $$ s_n=\sum_{k=-n}^{n}a_k\delta^{(k)}(x-k)$$ has a compact support $\Rightarrow s_n$ has finite order.

What should I do next?

GIFT
  • 321
  • By definition you need to show that the sequence $s_n(\phi)$ is convergent for every $\phi\in D$. This is clear since $\phi$ has compact support. – David C. Ullrich May 20 '19 at 18:40
  • @DavidC.Ullrich by definition you mean that I should write $\int s_n\cdot\phi dx $? – GIFT May 21 '19 at 09:36
  • No. By definition an element of $D'$ is a bounded linear functional on $D$; in particular $S:D\to\Bbb C$. The value $s(\phi)$ is not in general an integral. – David C. Ullrich May 21 '19 at 14:28
  • @DavidC.Ullrich I can't understand the evidence....I've been sitting for two days. Also I can't understand what does $\phi$ mean – GIFT May 24 '19 at 00:06
  • should I say that $s_n->0$ for every n? – GIFT May 24 '19 at 00:13
  • (i) I don't know what you mean when you say you can't understand what $\phi$ means. What's the definition of $D'$??? (ii) No, you shouldn't say $s_n\to0$ for every $n$; that makes no sense. – David C. Ullrich May 24 '19 at 13:53
  • @DavidC.Ullrich The definition of $D'$ I understand. This is the space of generalized functions, where generalized function is a continuous functional. Also it's written how $(f,\phi)$, where $\phi$ is a continuous function. – GIFT May 25 '19 at 11:05
  • I can't understand what definition I need to use.... – GIFT May 25 '19 at 11:06
  • No, a distribution is a continuous linear functional on $D$, the space of infinitely diifferentiable functions with compact support. We write $<f,\phi>$ for $f\in D$, not for continuous $f$!. Next: What's the definition of "$s_n\to s$ in $D'$"? That's the definition you need to use, since that's what you're trying to prove. – David C. Ullrich May 25 '19 at 12:45

0 Answers0