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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Fourier transform of a Radon measure

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 ...
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Does the Ramanujan-Petersson condition correspond to a Fourier type property?

The Ramanujan-Petersson is one of the requirements used in Selberg's class of L-functions, and as such is a necessary condition for the Riemann Hypothesis to hold. The general converse conjecture ...
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de Rham theorem for tempered distributions

I am wondering if the following statement holds. If $u\in \mathscr{S}'$ satisfies $\left< u,\Phi\right>=0$ for all $\Phi \in \mathscr{S}$ with $\mathrm{div}\, \Phi=0$, then there exists $p\in \...
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Looking for an electronic copy of Trèves and Martineau books

I would like to know if anyone has an electronic copy of the following books: MR0177291 Martineau, A.; Trèves, F. Éléments de la théorie des espaces vectoriels topologiques et des distributions. Fasc....
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Radon transform of the function $h(x_1,\ldots,x_n) = x_1 g(x_1,\ldots,x_n)$, where $g$ is the density of multivariate Gaussian $N(\mu,\Sigma)$

Given an absolutely integrable function $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined for every $(w,b) \in (\mathbb R^n \setminus \{0\}) \times \mathbb R$ by $$ R[f](w,b) := ...
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Restore initial condition for distributions

I am a bit confused by the following question and I hope someone could help me out. Let $u$ be the solution of the following initial value problem $$ u''(t) = g(t) \; \text{ in } (0,\infty), \quad\...
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Convergence in law and distribution theory

A standard result in probability theory asserts that a sequence of probablity measures $\mu_n$ on the Borel $\sigma$-algebra of $\bf R$ converges in law or weakly to a probability measure $\mu$ if and ...
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Functions of moderate increase compactly generated?

Let $\mathcal{O}_M(\mathbb{R}^d)$ be the space of smooth moderately increasing functions $\{ f \in \mathcal{C}^\infty(\mathbb{R}^d) : \forall \alpha \exists N \text{ such that} \Vert \langle \cdot \...
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Real life applications of distributions through models or simulations [closed]

What are the areas we can apply distributions in classical harmonic analysis? I don't mean probability distributions but distributions that are continuous linear functionals on the space of test ...
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$\frac{\partial f}{\partial \overline{z}}=0$ in distributional sense implies $f$ is holomorphic

Let $f=(u,v)\in \mathscr{D}'(U,\mathbb{C})$ be a distribution, where $U\subset\mathbb{C}=\mathbb{R}^2$ is an open set and $u$ and $v$ are the projection of $f$ onto the real and imaginary axis (ie $\...
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Anti-delta function?

Did anyone ever consider a "function" or "distribution" $F(x)$ with the following property: its integral $\int_a^b F(x)\,dx=0$ for any finite interval $(a,b)$ but $\int_{-\infty}^\...
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Have there been recent developments of Booker's approach to L-functions as distributions?

Andrew Booker introduced a framework to study L-functions through distributions in https://arxiv.org/abs/1308.3067v2. This allowed him and others to get new results about zeros of automorphic L-...
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Early successes of Schwartz distribution theory

What are the early successes of Schwartz distributions theory? What are the hard theorems that became simple and what open problems were solved with this new tool soon after Laurent Schwartz released ...
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Is there a characterization of tempered functions whose Fourier transforms are also tempered functions?

A tempered function on $\mathbb{R}^n$ is a locally integrable function that is tempered as a distribution, i.e. $L^1_{loc}\cap\mathcal{S}'$ is the space of tempered functions. This MSE question asked ...
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Integrals of the type $\delta(g^{n})$ on $\mathrm{SU}(2)$

I posted this question previously to MathSE. However, I have still not solved it, so lets try to ask it here. When doing some calculations with spin-foam models for 3d quantum gravity for some ...
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How to show that, $ C_c ( \pi_{1}^{ \mathrm{top} } (X) ) $ is dense in $ C_0 (X/G) $?

Let $G$ be a locally compact, Hausdorff, second countable, topological group. Let $X$ be a locally compact, Hausdorff $G$-space. The action of $G$ on $X$ gives an action of $G$ on $ C_0 (X) $ by, $$ (...
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Regularity of fiber integration between complex analytic spaces

Let $f:X\rightarrow Y$ be a flat surjective morphism between reduced complex analytic spaces. Assume that $Y$ is locally irreducible (i.e. unibranch). We assume that $X$ (resp. $Y$) is pure-...
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de Rham currents/distributions on manifolds with boundaries

My main source for currents and distribution theory on manifolds in general is de Rham's Differentiable Manifolds. To recap, let $M$ be a smooth, $m$ dimensional real manifold without boundary. De ...
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Space derivative of Brownian local time

Let $\{L_x(t):(t,x)\in [0,T]\times\mathbb R\}$ be the local time of a Brownian motion $(B_t)_{t\in [0,T]}$, I know that the map $x\mapsto L_x(t)$ is $\alpha$-Hölder for $\alpha<1/2$ (uniformly in ...
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Smooth cut-off in homogeneous Besov space

Given a Littlewood-Paley decomposition $$1 = \chi(\xi) + \sum_{j \geq 0}\varphi(2^{-j} \xi), \quad \xi \in \mathbb R^n$$ where $\chi$ is smooth, supported on a ball, and $\varphi$ is smooth, supported ...
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How to rigorously differentiate the convolution of a distribution and a $L^2$ function?

I want to prove the following: (Here, $W^{2,2}$ is a Sobolev space as defined in Evans, chapter 5; $S$ is a Schwartz space; and if $A$ is a distribution and $a$ a function, then $\langle A, a\rangle$ ...
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Spectral theorem and diagonal expansion for self adjoint operators

Asked by a physicist: In physics we use the fact that, given a self adjoint operator $A$, every element in Hilbert space can be written as "generalized linear combination" of the eigenstates ...
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Does convergence of tempered distributions implies convergence in $\mathcal{S}(\mathbb{R}^4,\mathbb{R})/\mathcal{S}_{0}$?

We can define the following symmetric semi-definite positive bi-linear form on $\mathcal{S}(\mathbb{R}^{4},\mathbb{R})$ with values in $\mathbb{C}$, \begin{equation}\label{prodintespaciales} (h_{...
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25 votes
4 answers
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Why are distributions "tempered"?

Google N-Gram shows that both "tempered distribution" and "temperate distribution" are used in English, but the first version significantly prevails, and usage of the second term ...
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4 votes
3 answers
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Fourier transform of periodic distributions

Following M. Ruzhansky and V. Turunen's book Pseudo-Differential Operators and Symmetries, in Chapter 3, Definition 3.1.25 (page 304), the space of periodic distributions is defined as follows (...
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Does this operator have a continuous, localized eigenfunction with negative eigenvalue?

I am looking at a class of operators $$ L[f](x)=af_{xxxx}-bf_{xx}+\frac{d}{dx}(\delta(x)f_x) $$ , a<0,b<0, on the real line, where $\delta$ is Dirac-delta. I am interested in ruling out the ...
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Evolution PDE in dual space : Generalization of a result of Gelfand

The following result is proved in "Generalized functions", Volume 3, Chapter 2.2 by Gelfand : Let $\Phi$ be a Fréchet space with dual space $\Phi'$ endowed with the weak topology. For ...
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How to prove that this one-parameter family of distributions converges to the Dirac measure?

While trying to understand a proof in a paper, I came upon the following a calculation needing the following identity: $$\lim_{t\to 0} \int_{-\infty}^\infty \left(e^{-\log(4\pi i t)/2} e^{ik^2/4t} -\...
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Approximating a subclass of $L^2(\mathbb{R})$ by Schwartz functions within similar subclass

It is well-known that real valued Schwartz functions on $\mathbb{R}_+$ $\mathcal{S}(\mathbb{R}_+)$ are dense in the set of square integrable functions on $\mathbb{R}_+$ $L^2(\mathbb{R}_+)$. We can ...
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3 votes
1 answer
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Is every compactly embedded Banach subspace of the space of distributions contained in some negative Holder space?

Let $F(=(C_c^\infty)^\ast,S^\ast)$ be the space of (tempered) distributions. Let $B\hookrightarrow F$ be a compactly embedded Banach subspace. Is it true that $B\subset C^{-\alpha}$ for some $\alpha&...
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6 votes
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Does $f \in L^1([0,T]; S'(\mathbb R^n))$ define a $(1+n)$-dimensional distribution?

Let $f : [0,T] \rightarrow S'(\mathbb R^n)$ be a family of tempered distributions satisfying $$\langle f(t), \phi \rangle \in L^1([0,T])$$ for any Schwartz function $\phi \in S(\mathbb R^n)$. Does $f$ ...
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Characterization of tempered distributions from tempered sequences

Let $\mathcal{D}(\mathbb{R})$ be the space of compactly supported and infinitely smooth functions for its usual topology. Let $\mathcal{D}'(\mathbb{R})$ be the topological dual of $\mathcal{D}(\...
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If theorem valid for compactly supported distribution then is it also valid for tempered distribution?

I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution. For instance, Theorem: Any $A \in \Psi^{m}$ ...
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2 votes
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Convergence in $S'(\mathbb R^d)$ of the paraproduct $\dot{T}_uv$

Let $B = B(0,4/3)$, $C = \{x \in \mathbb R^d : 3/4 \leq \|x\|_2 \leq 8/3\}$ and $\tilde{C} = \{x \in \mathbb R^d : 1/12 \leq \|x\|_2 \leq 10/3\}$. For a fixed Littlewood-Paley decomposition $\chi \in \...
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6 votes
2 answers
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A smooth function such that the second derivative of its absolute value is a distribution of positive order

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 ...
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6 votes
1 answer
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Is the tensor product of distributions a continuous bilinear map with respect to the weak topology?

Let $X$ and $Y$ be smooth manifolds. The map $\mathcal{D}'(X)\times\mathcal{D}'(Y)\to\mathcal{D}'(X\times Y)$ given by $(S,T)\mapsto S\boxtimes T$ is continuous with respect to the strong topology. Is ...
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16 votes
2 answers
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How to generalize the various vector calculus theorems to distributions?

Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...
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"Potential" for a divergence-free distribution

Edit: I would like to reopen this question since the linked potential duplicate question is not useful in showing that we get a tempered distribution as the potential, and I have found no easy way to ...
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4 votes
1 answer
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Uniqueness of distributional solutions to the Poisson equation

Let $S(\mathbb R^n)$ denote the space of all Schwartz functions on $\mathbb R^n$ equipped with the topology induced by the usual Schwartz semi-norms. Let $S(\mathbb R^n)^*$ denote its dual. My ...
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Derivation in Sobolev space [closed]

Let $f\in W^{1,\infty}(]0,T[)$ $(0<T\le\infty)$ such that $f(x)>0$ a.e. $x\in\mathopen]0,T[$ and let $$g(x)=e^{-\int_0^x \frac{ds}{f(s)}}$$ Formally $g' = -\frac{1}{f}g$. How can I justify this ...
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1 vote
1 answer
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How to understand subharmonic functions, distributions, and measure?

Sorry if this turns out to be a silly question, but I am having difficulties in both understanding it and finding other references for it. I hope that someone can clear my concepts here on overflow. ...
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5 votes
1 answer
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English translation of Schwartz's papers on vector-valued distributions

I am interested in systematically studying the theory of vector-valued distributions. The original two papers due to Laurent Schwartz entitled Théorie des distributions à valeurs vectorielles. I & ...
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Is this Beppo-Levi curl space a Banach space?

Let us define the quotient space: $$ V = \{ \mathbf{u} \in L^2_{loc}(\mathbb{R}^3; \mathbb{R}^3) : \operatorname{curl} \mathbf u \in L^2(\mathbb{R}^3; \mathbb{R}^3) \} / \nabla H^1_{loc}(\mathbb{R}^3)....
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1 answer
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Is delta function symmetric against real axis? [closed]

Is $\delta\left(a+bi\right)=\delta\left(a-bi\right)$? I wonder whether Dirac Delta (as defined via Fourier transform) is symmetric against the real axis. We can write Delta function as $$\delta(z) = \...
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4 votes
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Literature on the product of two distributions satisfying the Hörmander condition

I am currently studying some basic questions concerning the product $uv\in \mathscr{D}'(\mathbb R^n)$ of two Schwartz distributions $u,v\in \mathscr{D}'(\mathbb R^n)$ satisfying the Hörmander ...
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Poincaré's Lemma in the space of tempered distributions

It is well known that if $f\in \mathcal{D}'(\mathbb{R}^3,\mathbb{R}^3)$ and $\textbf{curl} f= 0$ then there exists a $u\in \mathcal{D}'(\mathbb{R}^3)$ such that $\nabla u = f$. Question. Does the ...
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Fourier Transform ; half space elliptic baby problem

I am attempting to look at some Liouville type theorems via a Fourier analysis approach and after looking at a baby problem I seem to be very confused. I assume this doesn't count as a research ...
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4 votes
1 answer
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Approximating compactly supported $L^2$ functions with Schwartz functions "from within"?

It is well known that the class of Schwartz functions $\mathcal{S}$ in dense in all $L^p$ spaces therefore for each $f \in L^2$ there exists a sequence of Schwartz functions $(f_k)$ such that $\lVert ...
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5 votes
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Feynman path integral and Wilsonian renormalization

Everything below is to be viewed in the Euclidean setting with $d$ dimensions and all measures are understood to be Borel measures. The usual problem of Quantum Field Theory is to make sense of ...
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Support of a fundamental solution of wave equation

The solution of the wave equation $$ \Box E = \delta $$ is $$ E(t,x) = \mathscr{F}^{-1} \left( \frac{\sin (t\lvert \cdot \rvert ) }{\lvert \cdot \rvert} \theta (t) \right)(x)\in\mathcal{S'}(\mathbb{R^{...
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