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 was brought up by the work of S. Sobolev and L. Schwartz in the middle of the $20^{th}$ century.

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2
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1answer
49 views

Prove that a given distribution is tempered

Suppose I have a distribution $E$ such that $\phi \ast E$ is square-integrable for all $\phi \in C_c^\infty \left( \mathbb{R}^d \right)$. Is it possible to prove that $E$ is tempered? It seems ...
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0answers
131 views

Does this formula correspond to a series representation of the Dirac delta function $\delta(x)$?

Consider the following formula which defines a piece-wise function which I believe corresponds to a series representation for the Dirac delta function $\delta(x)$. The parameter $f$ is the evaluation ...
3
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1answer
112 views

Distribution boundary value of analytic function and wave front sets

Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see ...
3
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0answers
108 views

Functional derivative as a Tensor of Dirac deltas

I'm working on a problem and I'm stuck at something that seems easy to solve but I couldn't get the solution so far. Let me introduce some concepts. [Pairings] Let $E$ and $F$ be Banach spaces over $...
4
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2answers
136 views

Function of moderate growth: history, motivation, and uses

I recently came across functions of moderate growth via this post and I was wondering, what are some concrete uses or applications of this space? Where does it appear and why was it introduced ...
6
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1answer
103 views

$GL_1(\mathcal{E}'(\mathbb{R}))$ open in $\mathcal{E}'(\mathbb{R})$?

Let $\mathcal{E}'(\mathbb{R})$ be algebra of all compactly supported distributions on $\mathbb{R}$, equipped with the strong dual topology $\beta(\mathcal{E}',\mathcal{E})$, and with the usual ...
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1answer
128 views

Pointwise functional derivative as partial derivative

Suppose $x_{1},...,x_{n} \in \mathbb{R}^{d}$ are fixed and $f: \mathcal{S}(\mathbb{R}^{d}) \to \mathbb{C}$ is given by: $$ f(\phi) = e^{\sum_{j=1}^{n}\alpha_{j}\phi(x_{j})}$$ with $\alpha_{1},...,\...
2
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3answers
342 views

Integral representation of tempered distributions

After my previous post I got curious about the following very simple question (which I don't seem to find the answer). Given a tempered distribution $K \in \mathcal{S}'(\mathbb{R}^{n_{1}+\cdots+n_{N}})...
2
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2answers
97 views

Representation of a Schwartz map in terms of a kernel

Suppose $f: \mathcal{S}(\mathbb{R}^{d})^{n+1} \to \mathbb{C}$ is a continuous function. To each $\varphi \in \mathcal{S}(\mathbb{R}^{d})$, we can define the map $f[\varphi]: \mathcal{S}(\mathbb{R}^{d})...
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A question about homogeneous distribution

A distribution in $\mathscr{S}^{\prime}\left(\mathbb{R}^{n}\right)$ is called homogeneous of degree $\gamma \in \mathbb{C}$ if for all $\lambda>0$ and for all $\varphi \in \mathscr{S}\left(\mathbb{...
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2answers
126 views

distributional divergence of the gravitational / Coulomb force close to the boundary

First of all, I am not sure of the terminology here, I am interested in the function $$F(x)=x|x|^{-d},x\in \mathbb{R}^d\setminus \{0\}$$ in dimension $d\geq 2$. I read somewhere that this is called ...
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1answer
124 views

Smoothness of family of distributions

Let $X$ be a compact manifold. Denote by $\mathscr{D}^\prime(X \times X)$ the space of tempered distributions on the cartesian product $X \times X$. Given two test functions $\varphi, \psi \in \...
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0answers
46 views

Less regular version of the Gaussian free field

One can define (continuous) Gaussian free field as follows: one can consider some orthonormal basis $(\psi_k)_{k=1}^{\infty}$ in the Sobolev space $H^1(\Omega)$ (here $\Omega \subset \mathbb{R}^d$) ...
3
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1answer
164 views

Mathematical meaning for the (continuous) Sine-Gordon transformation

I've been trying to understand the so-called Sine-Gordon Transformation which occurs in both classical and quantum statistical mechanics. One of the most cited references on this topic seems to be ...
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1answer
204 views

Did anyone ever introduce an “oscillating unity”?

I wonder whether anyone ever tried to introduce an extension of real numbers by adding an element $\nu$ which would signify the behavior of the function $(-1)^x$ as $x$ goes to infinity? In other ...
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3answers
195 views

on compact support distributions [closed]

If $f$ a distribution with compact support then they exist $m$ and measures $f_\beta$,$|\beta|\leq m$ such that $$f=\sum_{|\beta|\leq m}\frac{\partial^\beta f_\beta}{\partial x^\beta}$$ how to ...
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0answers
67 views

Structure theorem for distributions with support in a variety

Let $p\in \mathbb{C}[z_1,\cdots, z_d]$, and $V=\{{\mathbf{x}}\in \mathbb{R}^d: p({\mathbf{x}})=0\}$. Let $T\in \mathcal{D}'(\mathbb{R}^d)$ be a distribution whose support is contained in $V$. Is ...
3
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1answer
59 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 $n$ ...
4
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1answer
85 views

Convergence in $\sigma(\mathcal{E}',\mathcal{E})$ versus $\beta(\mathcal{E}',\mathcal{E})$

Let $\mathcal{E}'(\mathbb{R})$ be the space of all compactly supported distributions on $\mathbb{R}$. Suppose that $(T_n)$ is a sequence in $\mathcal{E}'(\mathbb{R})$ that converges to $T$ in the weak ...
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2answers
193 views

Derivatives of delta function as a basis for distributions [closed]

Is there some sense in which one could write any distribution as a sum of this sort? $$A(x,y)=\sum_{n=0}^{\infty}a_n(x)i^n\frac{\partial^n}{\partial x^n}\delta (x-y)$$ Provided that the rhs acting ...
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1answer
287 views

Kernel of the composition of operators

Let $X \subset \mathbb{R}^{n}$, $Y \subset \mathbb{R}^{m}$, and $Z \subset \mathbb{R}^{p}$ be open subsets, and let $K_P \in C_0^\infty(X \times Y)$ and $K_Q \in C_0^{\infty}(Y \times Z)$. Then, $K_P$ ...
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0answers
43 views

S-familiy induced by an operator induces a Schwartz function

Let $T:S(\mathbb{R}^d)\to S(\mathbb{R}^d)$, a continuous linear operator, where $S(\mathbb{R}^d)$ is the Schwartz space. There is a result that guarantees that the family $F=\{\delta_s\circ T\}_{s\in\...
2
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1answer
125 views

Continuity of convolution on $\mathcal{D}'_+$

Let $\mathcal{D}'_+:=\{T\in \mathcal{D}'(\mathbb{R}): \textrm{supp}(T)\subset [0,\infty)\}$. Here $\mathcal{D}'(\mathbb{R})$ is the usual space of distributions on $\mathbb{R}$, equipped with the weak$...
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0answers
37 views

Time derivative in parabolic Hölder spaces

Let $\Omega$ be a regular open set in $\mathbb{R}^n$ and $T>0$. Let $C^{\frac{1+\alpha}{2};1+\alpha}([0,T]\times \overline{\Omega})$ be the space of functions $f$ which are $\frac{1+\alpha}{2}$-...
6
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2answers
182 views

For which tempered distributions is the fractional derivative well-defined?

Let $\gamma \geq 0$ and consider the fractional derivative operator defined in Fourier domain by $$\mathcal{F} \{\mathrm{D}^{\gamma} \varphi \} (\omega) = (\mathrm{i} \omega)^{\gamma} \mathcal{F}\{\...
6
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2answers
200 views

Fourier coefficients of a periodic distribution?

Let $\tau>0$, and let $T\in \mathcal{D}'(\mathbb{R})$ be a $\tau$-periodic distribution (that is, $ \langle T, \varphi(\cdot+\tau)\rangle= \langle T,\varphi\rangle $ for all $\varphi \in \...
6
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1answer
643 views

Research topics in Microlocal Analysis

Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution ...
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0answers
174 views

Is polar decomposition of a smooth map Sobolev?

Motivation: Let $\mathbb{D}^2$ be the closed unit disk. I am studying the "elastic energy" functional $E(f)=\int_{\mathbb{D}^2} \text{dist}^2(df,\text{SO}_2)$, where $f \in C^{\infty}(\mathbb{D}^2,\...
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0answers
132 views

Is the Fourier transform of a measurable function as a tempered distribution necessarily a complex Borel measure?

Let $u\in\mathcal{S}'(\mathbb{R}^n)$. Suppose that $u$ is also a measurable function on $\mathbb{R}^n$. Is it true that the Fourier transform $\hat{u}$ as a tempered distribution is always a complex ...
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0answers
68 views

Characterizing geometrically Schwartz Kernels of pseudodifferential operators on a compact manifold

Let $M$ be a compact smooth manifold without boundary. Define $\mathcal{P} \subset \mathcal{D}^{'}(M \times M)$ to be the smallest linear subspace of the space of distributions on the product which is:...
8
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4answers
323 views

Defining the value of a distribution at a point

Let $\omega\in D'(\mathbb R^n)$ be a distribution and $p\in \mathbb R^n$. If there is an open set $U\subset \mathbb R^n$ containing $p$ such that $\omega|_U$ is given by a continuous function $f\in C(...
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1answer
121 views

Interchanging Integration Order involving Fourier Transform

$$f(\omega,u):=\frac1{\omega+iu}$$ where $i$ is the imaginary unit number. We see that the integral of a Fourier transform $$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...
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0answers
107 views

Fourier inversion formula for compactly supported distributions

I know that the Fourier transform of a compactly support distribution $u\in \mathscr{E}'(\mathbb{R}^{n})$ is smooth and also satisfies $$ |\hat{u}(\xi)|\leqslant C_{N}(1+|\xi|)^N,\label{1}\tag{1} $$ ...
4
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2answers
176 views

Fourier transform of a Lorentz invariant generalized function

Consider on $\mathbb{R}^{n+1}$ the indefinite quadratic form defining the Minkowski metric $$B(p)=(p^0)^2-(p^1)^2-\dots-(p^n)^2.$$ Let $\mu$ be a generalized function on $\mathbb{R}^{n+1}$ which is ...
5
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0answers
107 views

Extension of Valdivia-Vogt isomorphism from $\mathscr{D}(K)$ to $\mathscr{E}'(K)$

Let $M$ be a $d$-dimensional (say, Hausdorff, paracompact, connected and oriented) smooth manifold, and $K\subset M$ compact with $\mathring{K}\neq\varnothing$. M. Valdivia has shown (based on ...
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0answers
86 views

Entire analytic functions with entire analytic Fourier transform, and corresponding distributions

I'm interested in the Fourier transform on a space of distributions that includes more than the usual tempered distributions, and in particular allows for $\delta$-distributions supported at complex ...
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2answers
92 views

Delta-distribution composed with a function from the Fourier representation

A well known representation of Dirac's delta-distribution is via the Fourier transform of distributions: \begin{equation} \delta[f]:=f(0)=\int_{\mathbb{R}}\int_{\mathbb{R}} e^{\mathrm{i}xk}f(k)\mathrm{...
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1answer
182 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\...
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0answers
54 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
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1answer
100 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 $\mathcal{E}'(\mathbb{R})$?
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2answers
327 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 ...
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1answer
59 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)_{...
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0answers
60 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 $D$ ...
7
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2answers
251 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 ...
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2answers
170 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 ...
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0answers
134 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 ...
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1answer
198 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 ...
3
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1answer
1k 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
158 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 ...
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2answers
456 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 ...

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