**3**

votes

**0**answers

93 views

### Differential operators acting on the Schwartz space

I asked a similar question on math stack exchange but didn't get an answer so I will try to ask it here. Any help/suggestion is most than welcome!
Let $D$ be a linear differential operator with ...

**2**

votes

**0**answers

62 views

### Combining oscillatory integrals of the first and second kind

Consider an oscillatory integral of the first kind
$$
I_\lambda(x)=\intop_{\mathbb{R}^{n}}e^{i\lambda\Phi(x,y)}a(x,y)\,d y,\quad \lambda\geq 0,\; a\in C_c^\infty(\mathbb{R}^{k+n}),\; \Phi\in ...

**4**

votes

**1**answer

115 views

### normal form of currents?

(this question did not get any answers on math.SE, so I am reposting it here)
Let $M$ be an $n$-dimensional manifold. Then the space of currents $\mathcal D^k(M)$ of degree $k$ on $M$ is the space ...

**5**

votes

**3**answers

250 views

### A space of distributions vanishing on the boundary

The revised question
After more reflection on the problem, I might have found the answer by myself. Let $U$ be an open subset of $M$, irrespective of whether it has a boundary or not. Let
$$\mathcal ...

**1**

vote

**0**answers

97 views

### The dual of the space of smooth functions that vanish at infinity

Let $U \subset \Bbb R ^n$ be an open subset and let $\mathcal C$ be the space of the smooth functions on $U$ that vanish at infinity, endowed with the seminorms $p_\alpha (f) = \sup \limits _{x \in U} ...

**3**

votes

**1**answer

61 views

### Spaces $C^\infty(\mathbb T^n\times \mathbb R^n)$, $C^\infty_0(\mathbb T^n\times \mathbb R^n)$ and $\mathscr{S}(\mathbb T^n\times \mathbb R^n)$? [closed]

Is there any characterization of the space $C^\infty(\mathbb T^n\times \mathbb R^n)$ that I can take as a definition of it?
I assume it would be something like this:
$$C^\infty(\mathbb T^n\times ...

**3**

votes

**1**answer

310 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 ...

**7**

votes

**0**answers

175 views

### Wavelet-like Schauder basis for standard spaces of test functions?

The Schwartz space of test functions $\mathcal{S}(\mathbb{R})$ is isomorphic to $\mathfrak{s}$ the space of sequences of real numbers with faster than power-like decay. Likewise, the space ...

**4**

votes

**2**answers

287 views

### The derivative of a non-tempered distribution can be tempered?

Suppose we have a non- tempered distribution $u\in \mathcal D'(\mathbb R^d)\backslash \mathcal S'(\mathbb R^d)$. Is it possible to have $\partial_{x_1}...\partial_{x_d}u \in \mathcal S'(\mathbb R^d)$ ...

**4**

votes

**1**answer

137 views

### Hyperfunctions supported at a point

Is it true that the space of hyperfunctions on $\mathbb{R}^n$ supported at 0 coincides with the space of Schwartz distributions supported at 0?
More explicitly, is it true that any hyperfunction ...

**4**

votes

**0**answers

59 views

### Applications and main properties of hyperfunctions

I am trying to get familiar with hyperfunctions, and I do have some familiarity with the classical theory of distributions.
I am wondering whether hyperfunctions have any advantages over ...

**1**

vote

**2**answers

121 views

### Is there a version of the Titchmarsh Convolution theorem to find singular support?

Okay, some terminology, correct me if I'm wrong.
Singular support - the set on which a distribution fails to be smooth. In this case a piecewise function.
Is there a name for $f*f*f$? The ...

**1**

vote

**0**answers

77 views

### Construct a PDE solution from a net of approximations

Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$.
Let ...

**2**

votes

**1**answer

115 views

### Conformally covariant distributions

In Conformal Field Theory (in $D$ dimensions) one considers (in particular) correlation functions of the form
$$
\langle O(x)O(y)\rangle,
$$
where $O$ is a scalar primary field. Scale covariance ...

**5**

votes

**2**answers

236 views

### Recognizing Schwartz regular distributions

Are there characterizations of Schwartz regular distributions other than being locally integrable (which does not lend itself to easy manipulations)?
To be more detailed: if I want to show that some ...

**0**

votes

**1**answer

76 views

### The hypoellipticity of a heat-like operator

I am aware that the heat operator (on a smooth manifold) is hypoelliptic. I am also aware that there are manifolds on which the Schrödinger's operator (with a $\Bbb i = \sqrt {-1}$ multiplying $\frac ...

**3**

votes

**1**answer

101 views

### Generalized functions on a product of two manifolds

Let $X,Y$ be smooth compact manifolds. Let $C^\infty(X)$ and $C^{-\infty}(X)$ denote the spaces of smooth and generalized functions on $X$ respectively. We have the obvious canonical linear map ...

**0**

votes

**1**answer

142 views

### About weak derivatives [closed]

I have a question about weak derivatives.
Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some
open set $\emptyset \neq U \in \mathbb{R}^{n}$. We often say that $v$ is ...

**3**

votes

**1**answer

128 views

### Topologies on spaces of distributions and test functions

Let $X$ be an open subset of $\mathbb{R}^n$. Following the notation of Schwartz, we denote $\mathcal{D}$ the space of compactly supported complex-valued smooth functions on $X$ equipped with the ...

**5**

votes

**2**answers

276 views

### Weak solutions for a PDE of fourth order

I deal with two-dimensional Kirchhoff equation with $L^\infty$ coefficient and distributional right hand side:
$$
\Delta\Delta w+u(x,y)\left(\alpha^2\frac{\partial w}{\partial ...

**5**

votes

**0**answers

254 views

### Squaring random Schwartz distributions

Let $\mu$ denote the centered Gaussian measure on $S'(\mathbb{R}^d)$ with covariance
$$
\mathbb{E}
[\phi(f)\phi(g)]=\int_{\mathbb{R}^d} \frac{\overline{\widehat{f}(\xi)}
...

**5**

votes

**1**answer

221 views

### $\mathcal S(\mathbb R^n) \hat \otimes_\pi \mathcal S(\mathbb R^m) \simeq \mathcal S(\mathbb R^{n+m})$?

If $S(\mathbb R^n)$ is the Scwartz space of smooth rapidly decaying functions equipped with the topology generated by the family of semi-norms
$$\mathcal N_p (\varphi)= \sum_{|\alpha|, |\beta| \leq p} ...

**4**

votes

**0**answers

305 views

### distributions on Lie groups and representations

Let $G$ be a Lie group and $\pi$ a continuous action on $V$, a Fréchet space.
This action induces a representation of the space of compactly supported functions, $C_c(G)$, with convolution as product ...

**5**

votes

**0**answers

102 views

### Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?

Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mu$ be a probability measure ...

**3**

votes

**1**answer

440 views

### Understanding Bruhat's notion of Schwartz function

I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwartz-Bruhat space ...

**4**

votes

**1**answer

245 views

### Calculus of variation

This is probably simple but I'm stuck somewhere. I am trying to solve the calculus of variation problem that arise in an applied field: $$\min_{f \in C^1} \int^1_0 \int^1_0 (x-y)^2f(x,y)dxdy$$ ...

**2**

votes

**1**answer

140 views

### Well-posedness of heat equation with distributional right hand side

The question is about well-posedness of heat equation
$$
\frac{\partial\Theta}{\partial t}=\alpha^2\Delta\Theta+p(t)\delta(x-u(t))\delta(y-v(t)),~~ (x,y,t)\in\Omega\times[0,T],
$$
subjected to ...

**9**

votes

**4**answers

2k views

### Topology on the space of Schwartz Distributions

If we equip the Schwartz space $\mathcal{S}$ with its usual Fréchet space topology, then the space of continuous linear functionals $\mathcal{S}^\ast$ is known as the space of Schwartz distributions ...

**0**

votes

**2**answers

358 views

### Eigenfunction of local fractional derivative

Let $E_{\alpha}(x^{\alpha})$ be a Mittag-Leffler function, $\alpha \in (0,1)$. It is an eigenfunction for nonlocal fractional derivative, defined as a convolution with
$$
\Phi_{\lambda}(x) = ...

**5**

votes

**2**answers

248 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 ...

**3**

votes

**1**answer

184 views

### Extension of pseudodifferential operators

I'm very sorry if this is the wrong place to ask this question, but I've asked it on StackExchange and received no answers. ( ...

**7**

votes

**3**answers

1k views

### Rationale for Hadamard's finite part of a divergent integral

(Note: I asked this question a few days ago on math.stackexchange but didn't get any responses. I've therefore decided to post it here instead.)
I have a problem justifying throwing away the ...

**4**

votes

**2**answers

450 views

### Decomposition of distributions

Can we write every (tempered) distribution $\psi$, say on $\mathbb{R}$, as the sum of two distributions
$\psi = \psi_1 + \psi_2$
such that $\psi_1$ and the Fourier transform of $\psi_2$ are ...

**3**

votes

**1**answer

271 views

### Can I approximate Schwartz functions which integrate to zero by $C_0^\infty$ functions which integrate to zero?

Let $X$ be the closed subspace of Schwartz space $\mathcal{S}(\mathbb{R}^N)$ defined by
\begin{equation*}
X=\left\{f\in\mathcal{S}(\mathbb{R}^N):\quad \int f\; dx=0\right\}.
\end{equation*}
My ...

**5**

votes

**1**answer

198 views

### Are functions of moderate growth a bornological space?

I was thinking a bit about distribution theory the last weeks and stumbled across the following question:
There are two natural locally convex topologies on the space of smooth functions of moderate ...

**4**

votes

**0**answers

176 views

### Define the space of distributions with algebraic decay?

A tempered distribution $u\in \mathcal{S}'(\mathbb{R})$ is said to be rapidly decreasing if for every $f \in \mathcal{S}(\mathbb{R})$, $u*f \in \mathcal{S}(\mathbb{R})$.
One rough way to motivate ...

**0**

votes

**0**answers

51 views

### how can i solve this nonlinear problem?

let be the differential equation in dimension $ n > 3 $ or $ n =3 $
$$ -\Delta u =|grau|^{2}u $$ (1)
with the constraint that the vector 'u' $ |u|=1 $
then how could i prove that the unitary ...

**1**

vote

**0**answers

59 views

### Bounds on functions pullbacked via exponential map

Let us assume that $M$ is a compact Riemannian manifold (without boundary). For any point $x\in M$, we can pullback $C^\infty(M)$ functions to $T_x M$ via the exponential map, by setting
$$ (\exp_x^* ...

**3**

votes

**3**answers

342 views

### When sequentially continuous linear functional is continuous?

Let $C^\infty(X)$ denote the space of infinitely smooth functions on a compact manifold $X$ (at the beginning one may assume that $X$ is a circle, though I need a more general case). Let ...

**3**

votes

**2**answers

183 views

### Linear operators on distributions with different topologies

Denote by $\mathscr{D}^\prime$ and $\mathscr{D}^\prime_b$ the space of distributions on $\mathbb{R}^n$ equipped with the weak and the strong topology, respectively. Because the topology of ...

**7**

votes

**1**answer

356 views

### Is the space of rapidly decreasing (non-smooth) functions nuclear?

We denote by $\mathcal{S}(\mathbb{R})$ the space of smooth and rapidly decreasing functions. We define on $\mathcal{S}(\mathbb{R})$ the family of semi-norms
$$\lVert \varphi \lVert_{n,m} = \lVert ...

**4**

votes

**0**answers

147 views

### Convergence of Schwartz Kernels

I read this question, and I would like to ask the opposite: Assume that I have a sequence of smoothing operators $(P_n)$ with (hence smooth) kernels $(p_n)$ converging strongly to some smoothing ...

**3**

votes

**0**answers

165 views

### smoothing a current

Let $M$ be a smooth oriented manifold of dimension $n$ and $T$ a current of dimension $k$ on $M$. Let $\phi:P\times M \to M$ be a proper smooth family of diffeomorphisms of $M$ (i.e. $P$ is a smooth ...

**3**

votes

**1**answer

124 views

### Zero currents localized along a submanifold

Let $\mathcal{D}(\mathbb{R})$ be the continuous dual of $C^\infty_c(\mathbb{R})$, the space of compactly-supported smooth functions. There is a nice characterization of distributions ...

**6**

votes

**0**answers

128 views

### Norms and distributions

Question 1. Is there a nice or explicit way to describe the class of all distributions (generalized functions) $\mu$ on the $n$-sphere $S^n \subset \mathbb{R}^{n+1}$ for which the function
$$
F(v) := ...

**5**

votes

**3**answers

224 views

### Method to compute fundamental solutions which are distributions

The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on ...

**3**

votes

**0**answers

224 views

### Why distributions as functionals? [closed]

Why do we generalize functions by functionals on Schwartz Spaces, beyond the fact that it simply works? There should be a deeper reason why Schwartz considered functionals.
Excited for answers, Alex.

**1**

vote

**1**answer

76 views

### Estimate the analytical wavefront set $WF_A(u)$ given $WF_A(A_K u)$

Let $X$ and $Y$ be two real-analytic manifolds and $Z \subset X \times Y$ be a real-analytic embedded closed submanifold. Suppose now that $K \in \mathscr D'(X \times Y)$ is a distribution conormal to ...

**5**

votes

**1**answer

589 views

### Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$

Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$.
When $f$ is a function, it looks like the only solution is $f(x)=0$. But what if we allow distributions, such as the Dirac delta?

**4**

votes

**1**answer

168 views

### Practical way to check whether a distribution is conormal

Let $X$ be a $C^\infty$-manifold, $Y$ be its $C^\infty$-submanifold. Hörmander defines the set $I^m(X,Y)$ of conormal distributions as the set of all $u \in \mathscr D'(X)$ such that
$$
L_1 ...