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Results tagged with schwartz-distributions
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user 21907
A distribution is a continuous linear functional on the space $\mathcal{C}^{\infty}_c$ of smooth (indefinitely differentiable) functions with compact support. Though they appeared in formal computations in the physics and engineering literature in the late $19^{th}$ century, their formal setting was brought up by the work of S. Sobolev and L. Schwartz in the middle of the $20^{th}$ century.
2
votes
Accepted
Given a compact set $K \subset \mathbb{R}^n$, is the space of distributions supported on $K$...
A few reminders:
[1] The dual space of $\mathscr D(\mathbb R^n)=
C^\infty_c(\mathbb R^n)$ ($C^\infty$ functions with compact support) is
$\mathscr D'(\mathbb R^n)$ (distributions on $\mathbb R^n$).
[2 …
- 16.2k
3
votes
Singular support: equivalent definition
Let $U$ be an open subset of $\mathbb R^d$ and let $u\in \mathscr D'(U)$. Then we have
$$
(\text{supp } u)^c=\{x\in U, \exists V \text{open neighborhood of $x$ such that}\ u_{\vert V}=0\},
\tag{1}$$
…
- 16.2k
2
votes
Accepted
Microlocal approach to definition of product of distributions
Too long for a comment. For $u$ in $C^s$, $s\in (0,1)$, you can indeed define $u^2$ and then the distribution-derivative of $u^2$, which belongs to $B^{s-1}_{\infty,\infty}$. Now that does not define …
- 16.2k
0
votes
1
answer
304
views
Fourier transform of a Radon measure [closed]
Let $\mu$ be a Radon measure on $\mathbb R^d$
with finite total mass: I guess that it is a tempered distribution on $\mathbb R^d$ and thus one may consider its Fourier transform. Now I guess that its …
- 2,071
0
votes
Approximating a subclass of $L^2(\mathbb{R})$ by Schwartz functions within similar subclass
Let us follow the way that density is proven: obviously simple non-negative functions are dense (simple means finite linear combination of indicatrix functions of sets with finite measure) in non-nega …
- 16.2k
6
votes
2
answers
369
views
A smooth function such that the second derivative of its absolute value is a distribution of...
Let $f\in C^\infty(\mathbb R;\mathbb R)$ and let us define $g(x)=\vert f(x)\vert$. It is easy to verify that $g$ is locally Lipschitz-continuous function, but I would like to find an example of a smoo …
- 13k
3
votes
Accepted
Existence of a special function
From your assumptions, you have a $C^2$ function $\rho:\mathbb R^d\rightarrow \mathbb R$, such that
$$
D=\{x\in \mathbb R^d, \rho(x)<0\}, \quad \partial D=\{x\in \mathbb R^d, \rho(x)=0\},
$$
and
$
x\i …
- 16.2k
1
vote
Distribution boundary value of analytic function and wave front sets
You need to check Condition (ii) in Definition 8.2.2 in the first volume of Hörmander's ALPDO. Let us note $f(x+i0)$ the limit-distribution of your question and let $\Gamma$ be its wave-front-set. Let …
- 16.2k
7
votes
Accepted
Integral representation of tempered distributions
Let $\mathcal L$ be a continuous linear mapping from $\mathscr S(\mathbb R^n)$ into
$\mathscr S'(\mathbb R^n)$. The Laurent Schwartz kernel theorem asserts that there exists $K\in \mathscr S'( \mathb …
- 16.2k
1
vote
A question about homogeneous distribution
Let $u$ be a smooth function on $\mathbb R^n\backslash\{0\}$ homogeneous with degree $\lambda$ (on $\mathbb R^n\backslash\{0\}$). If $\lambda$ is not an integer $\le -n$, then $u$ can be uniquely exte …
- 16.2k
3
votes
1
answer
88
views
Division theorem for vector-valued distributions
The classical division theorem for scalar distributions on $\mathbb R^n$ can be formulated as follows. Let $T$ be a tempered distribution on $\mathbb R^n$ and let $P$ be a non-zero polynomial of $n$ …
- 16.2k
2
votes
Division theorem for vector-valued distributions
I think I have an answer to my own question : let us consider $Q$ the transposed of the comatrix of $P$. The determinant of $P$ is a polynomial and by the Lojasiewicz-Hörmander theorem, we can find a …
- 16.2k
3
votes
For which tempered distributions is the fractional derivative well-defined?
A preliminary remark.
The operator $(d/dx)^\gamma$ is never continuous on the Schwartz space (and thus on tempered distributions) except if $\gamma$ is a non-negative integer, since you introduce a s …
- 16.2k
8
votes
Were there attempts to express derivatives of Delta function as polynomials of Delta function?
There is a serious difficulty with the notion of products for distributions ; as a matter of fact Laurent Schwartz, one of the creator of Distribution Theory, wrote an article expressing the impossibi …
- 16.2k
3
votes
1
answer
109
views
A sufficient condition for a distribution to be temperate
Claim: Let $T$ be a distribution on $\mathbb R^n$ such that $\nabla T$ belongs to $L^p(\mathbb R^n)$ for some $p\in [1,+\infty]$. Then $T$ is a temperate distribution,
i.e. belongs to the topological …
- 91.8k