New answers tagged schwartz-distributions
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 ...
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 ...
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 ...
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 ...
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_{\...
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 ...
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 ...
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 ...
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