2
$\begingroup$

Let $\mathcal D$ be the space of 'test-functions'. Those are infinitely differentiable functions with compact support.

Define the following convergence on $\mathcal D$. $(\phi_j) \to \phi$ in $\mathcal D$ if:

  1. There exists a compactum $K$ that contains all the supports of $(\phi_j),j=1,2,\dots$ and $\phi$.

  2. For every multi-index $\alpha$ $(D^\alpha \phi_j)$ converges uniformly to $D^\alpha \phi$ on $K$.

Let now be $\phi \in D$ and $(v_n) \to 0$ in $\mathbb{R}$ and define $\phi_n(x)=\phi(x-v_n)$. I want to prove that $(\phi_n) \to \phi$ with respect to the convergence in $\mathcal D$. It seems quite obvious but I can't prove why. I started like this:

$|D^\alpha \phi_n - D^\alpha \phi | = |D^\alpha(\phi_n - \phi)|$ is bounded because $\phi_n - \phi$ is infinitely differentiable. Also $\phi_n - \phi \to 0$ pointwise and so $|D^\alpha(\phi_n - \phi)| \to 0$ put I can't see how uniform convergence follows.

$\endgroup$
  • $\begingroup$ Define translation operators $\tau_h$ per $\tau_h\varphi(x) = \varphi(x-h)$. Then $D^\alpha(\tau_h\varphi) = \tau_h(D^\alpha\varphi)$. Thus all you need to prove is $\tau_{v_n}\varphi \to \varphi$ uniformly on $\mathbb{R}^k$ for every continuous $\varphi$ with compact support, as $v_n\to 0$. $\endgroup$ – Daniel Fischer Jan 13 '14 at 15:38
0
$\begingroup$

Let $R$ such that the support of $\phi$ is contained in $[-R,R]$ and $\lVert v\rVert_\infty:=\sup_n|v_n|$. Then for each $n$, $\operatorname{supp}\phi_n\subset [-R-\lVert v\rVert_\infty,R+\lVert v\rVert_\infty]$ so 1. is fulfilled.

We have $$|\phi_n(x)-\phi(x)|\leqslant \int_{x-|v_n|}^{x+|v_n|}|\phi'(t)|\mathrm dt\leqslant 2|v_n|\sup_{t\in\mathbb R}|\phi'(t)|.$$ The same argument applies for the other derivatives.

$\endgroup$
  • $\begingroup$ Can you give some extra explanation on the first inequality? $\endgroup$ – Michiel Van Couwenberghe Jan 13 '14 at 15:51
  • $\begingroup$ And what happens for a function $\phi: \mathbb{R}^n \to \mathbb{R}$? $\endgroup$ – Michiel Van Couwenberghe Jan 13 '14 at 15:56
  • $\begingroup$ I used fundamental theorem of calculus. Then I bound the integral over $x-|v_n|$ and $x+|v_n|$ in order to deal with the case $v_n\leqslant 0$ and $v_n\lt 0$. For the multi-dimensional case, consider for a fixed $x$ and $n$ the map $t\mapsto \phi(x+tv_n)$ and look at its derivative. $\endgroup$ – Davide Giraudo Jan 13 '14 at 16:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.