3
votes
1answer
38 views

Identification of $L^2$ limits with distributional convergence

I just read the thread on "too much effort" and I would like to be more specific. Is the following reasoning correct: Let $g,g_\delta\in H^1(D)$, $D$ some domain in $\mathbb{R}^n$ with the following ...
0
votes
1answer
103 views

Convergence to $\delta$ distribution

Show that $$v_{t}(x) = (4 \pi kt)^{- \frac{1}{2}} \exp \left( -\frac{a x ^2}{4kt} \right)$$ converges to $\delta_{0}$ in $D'(\mathbb{R})$ when $t \to 0^{+}$. Asumming that: $$\int_{\mathbb{R}} ...
1
vote
1answer
118 views

A question on locally integrable function

Let $f$ be a locally integrable function, $f\in L_{\operatorname{loc}}^{1}(\mathbb{R}^n)$. Prove that the operator $$T_f:\phi\to\int_{\mathbb{R}^n}f(x)\phi(x)dx$$ is a distribution. (See ...
1
vote
0answers
60 views

Lebesgue-Stieltjes integral as a generalized function

Given some convex function $f(x)$, $x >0$ we can define a distribution $F \in \mathcal{D}'(0,\infty)$ using Lebesgue-Stieltjes integral $$ \langle F, \varphi \rangle ...