New answers tagged

4 votes
Accepted

Existence and uniqueness of solutions to a distributional ordinary differential equation

This second answer specifically addresses the issue of trying to interpret $v\circ x$ when $x$ is not regular and $v$ is a distribution. I will tie this into the edit to illustrate the problem. The ...
Willie Wong's user avatar
  • 37.4k
6 votes

Existence and uniqueness of solutions to a distributional ordinary differential equation

Let me explain several reasons why I think what you are trying to do cannot work. Algebra Suppose, for a moment, that $v(x(t))$ makes sense as a pull-back distribution. Then we would be allowed to ...
Willie Wong's user avatar
  • 37.4k
2 votes

Is there any work on distributional vector fields?

The specific question you ask here is basically about the existence of distributional antiderivatives. The answer is "yes", and that you can also recover the "$+C$". The main ...
Willie Wong's user avatar
  • 37.4k
1 vote

Is there any work on distributional vector fields?

Apart from signs, aren't you just asking for a function/distribution whose distributional derivative is $\delta$ (or $\delta+1$, or whatever)? Those integrals are by-abuse-of-notation just expressions ...
paul garrett's user avatar
  • 22.5k
3 votes
Accepted

Approximating a sequence of tempered distributions "uniformly" by Schwartz functions

For the first "highlighted issue": Not generally. For example, let $T$ be defined by $T\,g = \int_{-\infty}^{+\infty}g\,(s,s)\,{\rm d}s$. Taking $\eta$ with $\eta_1(0)=0$ we get $\lim_{\...
TaQ's user avatar
  • 3,348
5 votes
Accepted

For a tempered distribution $F$ on $\mathbb{R}^2$, what exactly does it mean by $\lvert F(x,y) \rvert \leq \lvert x-y \rvert^{-n}$?

I'll try to explain what Igor meant in his comments in a different way, maybe it helps. Of course, any tempered distribution is a distribution in the broader sense - more precisely, any compactly ...
Pedro Lauridsen Ribeiro's user avatar
4 votes

For a tempered distribution $F$ on $\mathbb{R}^2$, what exactly does it mean by $\lvert F(x,y) \rvert \leq \lvert x-y \rvert^{-n}$?

Distributions and ordinary functions are two different things, however, there are bridges between them, going both ways, modulo some technical hypotheses. If $f$ is an ordinary function one can ...
Abdelmalek Abdesselam's user avatar
4 votes
Accepted

Construction of random tempered distributions

Let me replace $\mathbb{R}_+\times \mathbb{R}^d$ by $\mathbb{R}$, the generalisation is an easy exercise. Write $\phi_n$ for the $n$th Hermite function, so that $\eta_n = \xi_{\phi_n}$ form a sequence ...
Martin Hairer's user avatar

Top 50 recent answers are included