Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [schwartz-distributions]

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 ...

1
vote
1answer
125 views

Convolution with Schwartz class function

Let $f, g\in \mathcal{S}(\mathbb R)$ (Schwartz class function), $\delta_0$ (dirac delta distribution). Consider distribution as follows: $$H(x, y)= f(x)g(x)\delta_0(y)-f(y)g(y)\delta_0(x), \ (x, y\...
0
votes
0answers
52 views

is this explicit linear operator hypo-elliptic

Consider an operator of the form $$L(\phi):=\Delta \phi + \gamma \phi_{rr}$$ here the $r$ denotes derivative with respect to the radial variable (we are in $ R^N$ say where $N \ge 3$). I am ...
4
votes
1answer
77 views

Is convolution jointly continuous on $\mathcal{E}'$?

Let $\mathcal{E}'(\mathbb{R})$ be equipped with its usual strong topology (being the dual space of $\mathcal{E}(\mathbb{R})$). Is convolution jointly continuous on $\mathcal{E}'(\mathbb{R})$?
9
votes
2answers
207 views

Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$

I asked this question on Math StackExchange, but it did not receive an answer, despite my offering a bounty to attract attention. I am unsure whether it is appropriate for this venue, but I thought ...
0
votes
1answer
50 views

Covergent net in $\mathcal{E}'(\mathbb{R})$ implies bounded?

Let $\mathcal{E}(\mathbb{R})$ be the space of all $C^\infty$ functions on $\mathbb{R}$ with its usual topology, and $\mathcal{E}'(\mathbb{R})$ be the dual space with the weak* topology. Let $(T_i)_{...
1
vote
0answers
45 views

Dense versus sequentially dense in $\mathcal{E}’$

Endow the dual space $\mathcal{E}’$ of smooth functions $C^\infty$ (with its metrizable topology described by uniform convergence on compacts for convergent sequences) with the weak* topology. Let $D$ ...
8
votes
2answers
188 views

On the Fourier-Laplace transform of compactly supported distributions

Let $\mathcal{E}'(\mathbb{R})$ be the space of all compactly supported distributions on $\mathbb{R}$. For $f\in \mathcal{E}'(\mathbb{R})$, let $\widehat{f}$ denote the entire extension of the ...
1
vote
2answers
153 views

A question arising in the distribution theory of L. Schwartz

Let $R$ be the ring of distributions $T\in \mathcal{D}'(\mathbb{R})$ with support in $[0,\infty)$ and with the operations of pointwise addition and multiplication taken as convolution, and $I$ be the ...
2
votes
0answers
119 views

Theory of distributions on various domains

The prototypical example of a distribution is the Dirac delta function, defined as a linear functional taking a well behaved test function $\phi:\mathbb{R} \to \mathbb{R}$ and returning its value at ...
1
vote
1answer
130 views

Schwartz distributions, Colombeau algebra and applications

I have studied "enough" the theory of distributions , I would like to deepen some topic with applications. With some research I arrived at this book: "Geometric Theory of Generalized Functions with ...
2
votes
1answer
213 views

About the Fourier transform of the logarithm function

I want to calculate / simplify: $$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$ where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\...
5
votes
1answer
126 views

The division problem for tempered functions

It is well known (see for example S Łojasiewicz, Sur le problème de la division, Studia Math. 8 (1959), 87–136.) that any linear partial differential operator with constant coefficients is surjective ...
-1
votes
2answers
290 views

Were there attempts to express derivatives of Delta function as polynomials of Delta function?

Is seems to me that it makes sense to presume some relations between derivatives of Dirac delta functions and its powers. I wonder, whether someone proposed a similar theory? Particularly, it could ...
5
votes
3answers
724 views

The “Spaces of Schwartz distributions are finite dimensional” challenge

The more I study Schwartz distributions and the corresponding spaces, the more the latter look "finite dimensional" to me. Of course they are not finite dimensional in the technical sense but they are ...
16
votes
3answers
1k views

Research topics in distribution theory

The theory of distributions is very interesting, and I have noticed that it has many applications especially with regard to PDEs. But what are the research topics in this theory? also in terms of ...
1
vote
0answers
51 views

Support of functions on Minkowski space and their Fourier transform

Suppose that a tempered distribution $f(x)$ on 4-dimensional Minkowski space with signature $+---$ vanishes for $x^2<0$ and its Fourier transform $\hat f(p)$ vanishes for $p^2<0$. Are then $f$ ...
3
votes
1answer
269 views

A Schwartz kernel type theorem for Sobolev spaces

The well-known Schwartz kernel theorem states that a continuous operator from smooth test functions to distributions, $T \colon C^\infty_c(\mathbb{R}^n) \to C^\infty_c(\mathbb{R}^n)'$ is continuous ...
4
votes
1answer
98 views

Neumann DBAR problem with tempered distributions

It is well-known that the operator $$\frac{\partial}{\partial \overline{z}} : C^{\infty}(\mathbb{C}) \to C^{\infty}(\mathbb{C})$$ is surjective. (And it also works if we replace functions by Schwartz ...
4
votes
2answers
436 views

Completion of $\mathcal{S}(\mathbb{R})$ for a given norm

Assume that $\lVert \cdot \rVert$ is a norm on the space of rapidly decaying functions $\mathcal{S}(\mathbb{R})$. Under which conditions on the norm can we say that the completion $\mathcal{X}$ for ...
2
votes
0answers
42 views

Generalized Besov spaces with different integrability and smoothness in space and time?

Consider the family of Besov spaces $B_{p,q}^{s}(\mathbb{R})$ with $0<p,q \leq \infty$ and $s \in \mathbb{R}$. Is there a natural way to define spaces of generalized functions $f(t,x) \in \mathcal{...
1
vote
0answers
39 views

Is $\mathscr{S}_h'$ a complementary subspace for $\mathscr{S}'/\mathscr{P}$, the space of tempered distributions modulo polynomials?

Recall that in many Fourier Analysis texts, given a function $\Psi$ such that $\hat{\Psi}\in\mathcal{D}(\mathbb R^d)$, $\hat\Psi\ge0$ is supported in an annulus, and $\sum_{j\in\mathbb Z}\hat\Psi(2^j\...
6
votes
1answer
267 views

Regularity of Fourier transforms of $L^p$ functions for $2<p\le\infty$

I was recently reading about the Mikhlin and Hörmander Multiplier Theorems, which give conditions for a measurable function $m:\mathbb R^d\to\mathbb C$ to be an $L^p$ multiplier, i.e. for there to ...
2
votes
0answers
75 views

Extension of a generalized function to the plane

Let $\phi$ be a generalized function on $\mathbb{R}^2\backslash\{0\}$, and assume that its differential $d\phi$ extends to the whole plane $\mathbb{R}^2$. Q. Does $\phi$ also extend to $\mathbb{R}^...
5
votes
1answer
174 views

Cohomology of sheaf of Schwartz distributions with support in a submanifold

Let $M$ be a smooth manifold. Let $Z\subset M$ be a smooth submanifold which is a closed subset. Let $F$ denote the sheaf of generalized functions (equivalently, Schwartz distributions) on $M$, namely ...
3
votes
1answer
84 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 ...
4
votes
2answers
211 views

Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold

I'm trying to close in on a definitive answer to my own question BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?, and think I ...
5
votes
1answer
278 views

The elliptic regularity theorem for differential operators with variable coefficients

I'm following the book "Introduction to the theory of distributions" by Friedlander and Joshi. There is the following result p. 109 Theorem (8.6.1). Let $X \subset \mathbb{R}^n$ be an open set, and ...
2
votes
2answers
284 views

Fourier transform inversion theorem for a function not in L1 or L2

For $\frac{1}{4}<a<1$ consider the following function: $$f(x)=\frac{|x|^{\frac{1}{2}}}{(x^2+1)^{a+ib}}$$ If $1>a>\frac{1}{2}$ then $f(x) \in L^2$ and the Fourier inversion theorem can be ...
3
votes
0answers
85 views

Question on de Rham complex with distributional coefficients

Let $X$ be a smooth manifold (usually assumed to be paracompact). Let us denote by $\underline{\Omega}^{p,-\infty}_X$ the sheaf of real valued $p$-forms with distributional coefficients in the ...
11
votes
1answer
545 views

Is every continuous endomorphism of the Schwartz space a pseudo-differential operator?

Let $\mathcal{S}:= \mathcal{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions with rapid decay. The question is pretty simply stated in the title. Pseudo-differential act continuously on the ...
5
votes
3answers
348 views

Exponential derivative of delta distribution?

This question is from here. I'm asking it here as well to increase the number of people who see it and might be able to help. The question is, what is the result of the following integral for integer ...
5
votes
0answers
83 views

Geometric characterization of Silva distributions

There is a well known geometric characterization of tempered distributions on $\mathbb{R}^n$. A distribution $T\in \mathcal{D}'(\mathbb{R}^n)$ is an element of $\mathcal{S}'(\mathbb{R}^n)$ if and ...
1
vote
0answers
128 views

Can we get rid of this test function?

I have a real-valued function $f$ defined on a ball $B$ of $\mathbb{R}^{N}$, $N\geq2$. I have found a constant $M>0$ such that for all $x\in B$ and $B(x,R)$ (ball of center $x$ and radius $R>...
4
votes
1answer
302 views

How to show that this limit converges in the distributional sense to a dirac delta function

Let $$\begin{eqnarray}\nonumber f(y, t) &=& \frac{C}{\sigma ^2 t} \left[\frac{(1-\alpha) (b-y)}{\alpha t^{\alpha}} \, _1F_1\left[\frac{\alpha+1}{2 \alpha};\frac{3}{2};-\frac{ (b-y)^2}{2 \...
3
votes
0answers
123 views

(Non-)Existence of certain invariant distributions on a p-adic space

Following Bernstein-Zelevinski, an $\ell$-space is a Hausdorff, locally compact totally disconnected topological space. For an $\ell$-space $X$, denote $S(X)$ the space of Bruhat-Schwartz functions on ...
0
votes
2answers
176 views

Is this equality between an integral and a series wrong?

In this paper (Maroun's PhD dissertation, 2013) at page 46 the following formula is given (apparently without a reference): $$\int_0^{\infty } e^{i a x^s+i b x^p} \, dx=\sum _{n=0}^{\infty } \frac{\...
1
vote
1answer
213 views

Fourier transform of delta function restricted to sphere [duplicate]

I want to compute $\mathcal{F}^{-1}\{\delta(|\cdot|-1)\}(x)$, which exactly means the following computation: $$f(x) = (2\pi)^{-n/2} \int_{|\xi|=1}e^{ix\cdot\xi}\mathrm{d}\xi, \mbox{ where }~ \xi \in \...
3
votes
1answer
71 views

Distributions on the mirabolic subgroup which are left invariant to the unipotent radical

I'm trying to find a reference or a proof to the following statement used by Matringe, N., 2012. Cuspidal representations of GL (n, F) distinguished by a maximal Levi subgroup, with F a non-...
1
vote
0answers
76 views

Is a smooth function with compact support defined on adele groups of Schwartz class?

I'm reading Gelbart's Introduction to the Selberg Trace Formula https://arxiv.org/abs/math/0407288. In his paper he seems to have used the consequence that a smooth function with compact support is a ...
5
votes
3answers
227 views

No kernel of the form $\lvert x - y\rvert^{-1}$ on tempered distributions?

Does there exist a continuous bilinear form $\mathcal{B}$ on $\mathcal{S}(\mathbb{R})\times \mathcal{S}(\mathbb{R})$ such that \begin{equation} \mathcal{B}(\varphi_1, \varphi_2) =\int_{\mathbb{R}\...
3
votes
1answer
198 views

Wave front set of vector-valued Dirac delta distribution

Context: I am reading a physics paper Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime which applies the notion of the wave front set to operator-valued ...
3
votes
0answers
137 views

Extension of Paley-Wiener-Schwartz theorem to vector-valued distributions

Let $H_{j} := (H_{j}, \| \cdot \|_{H_{j}} ), j=0,1$ be a Hilbert space, and set \begin{equation*} {\mathscr S}'(\mathbb{R}^{n}, H_0; H_1) := {\mathscr L}( {\mathscr S}(\mathbb{R}^{n}, H_0), H_1) \end{...
6
votes
0answers
176 views

Uniform estimates of Fourier transform of tempered functions with parameters

Consider the following function in $\mathbb{R}^3$: $$ f_t(x)=(1+|x|^2)^{-\alpha}e^{-g(x)t},\,\,\,\,\, \text{where}\,\, g(x)=\frac{x^2_1\cdot x^2_2}{1+|x|^2}, $$ where $\frac{1}{2}<\alpha<1$, and ...
3
votes
2answers
218 views

limit of a sequence of distributions

Let $u\in \mathcal{D}'(\mathbb{R}^d)$, $f\in C_c^\infty(\mathbb{R}^d)$ and $f(x)=1$ if $|x|\leq 1$; $f(x)=0$ if $|x|>2$. Can we get the following conclusion: there exists $v\in \mathcal{D'}(\mathbb{...
3
votes
1answer
211 views

Hörmander's hypoellipticity theorem for complex coefficients

Hörmander's theorem says that if $L = \sum _{i=1} ^r X_i ^2+ X_0 + f$ on some open subset $U \subseteq \Bbb R$ has the property that the Lie algebra generated by $\{X_0, \dots, X_r\}$ at every point ...
0
votes
0answers
27 views

Positivity of continuous functions with values in distributions

Let $f\in C([0,T],\mathcal{S}')$ and assume $f(0)$ is a positive distribution. Does there exist $0<\epsilon\le T$ such that $f$ remains positive on $(0,\epsilon)$? If not, what further assumptions ...
2
votes
1answer
141 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 ...
3
votes
2answers
261 views

Completeness of an exponential family

The question is this: Does there exist an integrable function $f\colon\mathbb R\to\mathbb R$ such that $f$ differs from $0$ on a set of nonzero Lebesgue measure and \begin{equation} \int_{\mathbb R}...
12
votes
1answer
605 views

On an Inequality of Lars Hörmander

Let $P(z)$ be a non-null complex polynomial in $\nu$ variables $z=(z_1,\dots,z_n)$ of degree $\mu$: \begin{equation} P(z)=\sum_{|\alpha| \leq \mu} c_{\alpha} z^{\alpha}, \end{equation} where as usual ...
2
votes
0answers
121 views

Does the reciprocal of a polynomial define a tempered distribution when it is locally integrable?

Consider a complex polynomial in $n$ variables $z=(z_1,\dots,z_n)$: \begin{equation} P(z)=\sum_{|\alpha| \leq N} c_{\alpha} z^{\alpha}, \end{equation} where as usual for every $\alpha=(\alpha_1,\dots,\...