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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.
18
votes
3
answers
7k
views
Eigenvectors of the Fourier transformation
The Fourier transform $\hat u$ is defined on the Schwartz space $\mathscr S(\mathbb R^n)$
by
$
\hat u(\xi)=\int e^{-2iπ x\cdot \xi} u(x) dx.
$
It is an isomorphism of $\mathscr S(\mathbb R^n)$ and the …
- 16.2k
11
votes
2
answers
701
views
Poincaré lemma for distributions
Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form
$$
u=\sum_{1\le j\le n} u_j dx_j,\quad …
- 16.2k
8
votes
Were there attempts to express derivatives of Delta function as polynomials of Delta function?
There is a serious difficulty with the notion of products for distributions ; as a matter of fact Laurent Schwartz, one of the creator of Distribution Theory, wrote an article expressing the impossibi …
- 16.2k
8
votes
Why is multiplication on the space of smooth functions with compact support continuous?
The spaces $C^\infty_c(\mathbb R^d)$ and $C^\infty_c(\mathbb R^d)\times C^\infty_c(\mathbb R^d)$ are $LF$ spaces (inductive limit of Frechet spaces) and their standard topologies are not metrizable. W …
- 16.2k
7
votes
Accepted
Integral representation of tempered distributions
Let $\mathcal L$ be a continuous linear mapping from $\mathscr S(\mathbb R^n)$ into
$\mathscr S'(\mathbb R^n)$. The Laurent Schwartz kernel theorem asserts that there exists $K\in \mathscr S'( \mathb …
- 16.2k
7
votes
2
answers
467
views
Eigenstates of Fourier transformation
Let $\gamma$ be defined on $\mathbb R^n$ by $\gamma (x)=e^{-π x^2}$. With $\mathcal F$ standing for the Fourier transformation defined on the Schwartz space by
$$
(\mathcal F u)(\xi)=\int e^{-2iπ x\cd …
- 16.2k
7
votes
1
answer
4k
views
The Schwartz space is not normable
The Schwartz space of rapidly decreasing function (as well as their derivatives) on $\mathbb R^n$ is a Fréchet space, whose (metric complete) topology is given by the usual countable family of semi-n …
- 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 …
- 16.2k
5
votes
Accepted
Practical way to check whether a distribution is conormal
Let us take a look at the case where $Y\equiv x_n=0$ in $\mathbb R^n$.
Let $a(\underbrace{\overbrace{x_1,\dots,x_{n-1}}^{x'},x_n}_x; \xi_n)$ be a symbol in $S^m(\mathbb R^n\times \mathbb R)$ and let …
- 16.2k
4
votes
Accepted
One-sided Cauchy principal value
Let me give an example: you want to define a distribution on $\mathbb R$ which coincides with $1/t$ on $(0,+\infty)$ and vanishes on $(-\infty,0)$. Let us take
$$
T=\frac{d}{dt}(H(t)\ln t),\quad H=1_{ …
- 16.2k
3
votes
Accepted
Extension of pseudodifferential operators
Your operator $A$ is continuous from $C_0^\infty(M)$ into $C^\infty(M)$, so the adjoint $A^*$
is continuous from $\mathcal E'(M)$ into $\mathscr D'(M)$. Now, the operator $A^*$ is also a pseudodiffere …
- 16.2k
3
votes
1
answer
322
views
Characterization of convex functions
Let $\Omega$ be a convex open subset of $\mathbb R^n$ and let $f:\Omega\rightarrow \mathbb R$ be a convex function. Since $f$ is continuous, it can be considered as a distribution on $\Omega$ and then …
- 16.2k
3
votes
Accepted
Why equality of singular supports?
$\nu$ is a distribution on the real line and the operator $P=d/dt$ is elliptic with constant coefficients. In that case we have
$$
\text{singsupp $\nu$}=\text{singsupp $P\nu$}
$$
for the $C^\infty$ si …
- 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 …
- 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