# Tagged Questions

Use this tag for questions about distributions (or generalized functions). For questions about "probability distributions", use (probability-distributions). For questions about distributions as sub-bundles of a vector bundle, use (differential-geometry).

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### Joint probability density function $(X^2,Y^2)$

Let $X$ and $Y$ be ramdom variables having the following joint probability density function $f(x,y)=\begin{cases} \frac{3}{8}xy & x\geq0,\,y\geq0,\:x+y\leq2\\ 0 & otherwise \end{cases}$ ...
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### If an integrable function is orthogonal to all derivatives, then is f a constant?

Suppose that I have a function in $f \in L^1(\mathbb{R})$ such that $$\int_{\mathbb{R}}f(x)v'(x)\,dx = 0$$ for all test functions $v$ which are smooth with compact support. Can I show that $f(x)$ is ...
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### Fundamental theorem of calculus for distributions

I want to show that if $f\in C([0,1])$ and the distributional derivative $f'$ on $(0,1)$ is in $L^1((0,1))$, then $$f(1) - f(0) = \int_0^1 f'(x)\,dx$$ I am having a lot of trouble getting started. ...
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### Can a schwartz class function be dominated by an exponential?

Given a function $f$ from Schwartz class, does there exist a constant $C$ such that $|f(x)|<Ce^{-|x|}$. For me its seems true, if it is not true, any counterexample would be very illustrative to me
Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the $L^2(\Omega)$- or $L^2(\Omega,\mathbb R^d)$-inner product (depending on the context) $\mathcal ... 1answer 28 views ### Variation of parameters for ODES with distributions as coefficients In my work, I encounter the following type of equation. Consider a non-homogeneous system $$X'=A(t)X+f(t),\;\;\;\;\;\;\;\;(1)$$ where$X$is a$n$-dimensional vector valued ... 1answer 57 views ### Has the distributional Laplacian$\Delta f:C_c^\infty(\Omega)'\to C_c^\infty(\Omega)'$a unique extension in$H_0^1(\Omega)'$? Let$d\in\mathbb N\Omega\subseteq\mathbb R^d$be open$\mathcal D:=C_c^\infty(\Omega)$and $$H=\overline{\mathcal D}^{\langle\;\cdot\;,\;\cdot\;\rangle_H}\tag 1$$ with $$\langle\phi,\psi\rangle_H:... 0answers 27 views ### Product of currents De Rham currents are for differential forms as distributions are to (smooth) functions. There is a notion of exterior product for differential forms: I wonder whether there is an algebra structure on ... 1answer 28 views ### Composition with Second Derivative of Dirac Distribution Here's a nasty question that came up on an old qualifying exam that I'm helping students study. Let T = \delta^{\prime\prime}(\cos(x)) and let \varphi = e^{-x^2}. Evaluate \langle T,\... 2answers 30 views ### how far the distribution from the uniform distribution I have two discrete probability distributions P and Q, where P=(p_1,...,p_n) and Q=(q_1,...,q_n), in addition I have uniform distribution U=(\frac{1}{n},...,\frac{1}{n}). The question is ... 4answers 52 views ### C_c^\infty(\Omega)\subseteq L^p(\Omega) for any open \Omega? Let d\in\mathbb N and \Omega\subseteq\mathbb R^d. Can we show that$$C_c^\infty(\Omega)\subseteq L^p(\Omega)\tag 1$$for all p\in [1,\infty]? It's clear that (1) holds if \Omega has finite ... 2answers 50 views ### If p,q are distributions with \partial_ip=\partial_iq, then p=q Let d\in\mathbb N \Omega\subseteq\mathbb R^d \mathcal D(\Omega):=C_c^\infty(\Omega) Let$$\frac{\partial p}{\partial x_i}(\phi):=-p\left(\frac{\partial \phi}{\partial x_i}\right)\;\;\;\... 0answers 29 views ### Deducing an equality involving Fourier transform of Schwarz functions For$\psi,\phi\in\mathscr{S}(\mathbb{R}^n)$we have $$\psi(0)\int_{\mathbb{R}^n}\mathscr{F}\phi(\xi) \, d\xi = \phi(0) \int_{\mathbb{R}^n} \mathscr{F}\psi(\xi) \, d\xi$$ I want to prove that this ... 1answer 38 views ### Volterra Operator on Sobolev Space I stumpled over the following result in a script: Let$1 \leq p < \infty$and$f \in L_p[a,b]$. Define the Volterra operator as $$Vf(t) = \int_a^t f(s) ds.$$ Then we have$Vf \in W^{1, p}[a,b]$... 2answers 44 views ### What is the divergence of a distribution? Let$d\in\mathbb N\Omega\subseteq\mathbb R^d$be open$\mathcal D(\Omega):=C_c^\infty(\Omega)$If$p\in \mathcal D'(\Omega)$, then $$\frac{\partial p}{\partial x_i}(\phi):=-p\left(\frac{\... 1answer 49 views ### Definition of the Laplacian as an operator from H_0^1(\Omega) to H_0^1(\Omega)' Let d\in\mathbb N \Omega\subseteq\mathbb R^d be open \mathcal D(\Omega):=C_c^\infty(\Omega) f\in L^2(\Omega) and$$\langle f\rangle:=\left.\langle\;\cdot\;,f\rangle_{L^2(\Omega)}\right|_{\... 1answer 43 views ### Null space of the Laplacian operator? (I guess the answer to my question is well-known in harmonic analysis, but I consider it in the framework of Schwartz distributions and in any dimension, and could not find a satisfactory answer.) ... 1answer 42 views ### Proving that a function belongs in the space of tempered distributions Let$a>0$and define $$g(\xi):=\frac{\sin a\xi}{\xi(1+\xi^{2})}$$ I want to prove that$g\in\mathscr{S}'(\mathbb{R})$and consquently that$g\in L^{1}(\mathbb{R})$(but this implication is ... 1answer 28 views ### Can we talk about the adjoint of a linear operator defined on a distribution space? Let$d\in\mathbb N\Omega\subseteq\mathbb R^d$be open$\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)\langle\;\cdot\;,\;\cdot\;\rangle$denote the inner product on$L^2(\Omega,\mathbb R^d)$and $$... 1answer 41 views ### Distributional limit of f_n = e^{x/n} Find the distributional limit of f_n = e^{x/n}. That means that I have to find the limit of T_{f_n}. I proceeded as follows. Let \phi \in \mathcal D be a test function. Then:$$\langle T_{f_n},... 1answer 48 views ### How to show that a Schwartz distribution is in a Lebesgue or Sobolev space? It is known that all$L^p$spaces (and, consequently, all$W^{s,p}$spaces) can be embedded in the space of Schwartz distributions$\mathcal D '$. There is a problem, though: how do I check whether ... 0answers 30 views ### Prove that$\left.F\right|_{\left\{ϕ∈C_c^∞(Ω,ℝ^d):∇⋅ϕ=0\right\}}=0⇔∃p∈C_c^∞(Ω)$with$F=∇p$, for all$F∈H_0^1(Ω,ℝ^d)'$Let$d\in\mathbb N\Omega\subseteq\mathbb R^d$be open$\langle\;\cdot\;,\;\cdot\;\rangle$denote the inner product on$L^2(\Omega,\mathbb R^d)\mathcal D:=C_c^\infty(\Omega,\mathbb R^d), H:=\... 1answer 40 views ### Property singular support of the convolution of distributions Let u \in \mathcal{D}'(\Omega) and U \subset \Omega open. By definition we say that u \in \mathcal{E}(U) if \exists u(x) \in \mathcal{E}(U) such that \begin{align*} \displaystyle \langle \... 2answers 63 views ### Relationship between C_c^\infty(\Omega,\mathbb R^d)' and H_0^1(\Omega,\mathbb R^d)' Let d\in\mathbb N \Omega\subseteq\mathbb R^d be open \langle\;\cdot\;,\;\cdot\;\rangle denote the inner product on L^2(\Omega,\mathbb R^d) \mathcal D:=C_c^\infty(\Omega,\mathbb R^d) and... 1answer 60 views ### Is the restrictionf$of$F\in H^{-1}(\Omega,\mathbb R^d)$to$C_c^\infty(\Omega,\mathbb R^d)$a distribution? Let$d\in\mathbb N\Omega\subseteq\mathbb R^d$be open$\langle\;\cdot\;,\;\cdot\;\rangle$denote the inner product on$L^2(\Omega,\mathbb R^d)\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$and $$... 0answers 101 views ### If F∈H^{-1}(Ω,ℝ^d) and ∃p∈\mathcal D'(Ω):\left.F\right|_{\mathcal D(Ω,ℝ^d)}=∇p, then ∃\overline p∈H^{-1}(Ω):F=∇\overline p Let d\in\mathbb N \Omega\subseteq\mathbb R^d be open \mathcal D(\Omega):=C_c^\infty(\Omega) and \mathcal D(\Omega,\mathbb R^d):=C_c^\infty(\Omega,\mathbb R^d) H^{-1}(\Omega):=H_0^1(\Omega)'... 0answers 27 views ### If p is a distribution such that \nabla p is regular, then p must be regular too Let d\in\mathbb N \Omega\subseteq\mathbb R^d be open \mathcal D(\Omega):=C_c^\infty(\Omega) q\ge 1 Each f\in L^1_{\text{loc}}(\Omega) can be identified with \langle f\rangle\in\mathcal ... 0answers 36 views ### I don't understand De Rham's theorem about the gradient of a distribution Let d\in\mathbb N \Omega\subseteq\mathbb R^d be open \mathcal D(\Omega):=C_c^\infty(\Omega) and$$\mathfrak D(\Omega):=\left\{\Phi\in\mathcal D(\Omega)^d:\nabla\cdot\Phi=0\right\}$$In a ... 0answers 22 views ### Characterization of the Gradient of a Distribution Let d\in\mathbb N \Omega\subseteq\mathbb R^d be open \mathcal D(\Omega):=C_c^\infty(\Omega) (without topology) u:\mathcal D(\Omega)\to\mathbb R is called distribution on \Omega :\... 0answers 15 views ### Characterization of a set occurring in the Helmholtz-Hodge decomposition Let d\in\mathbb N \Omega\subseteq\mathbb R^d be open \mathcal D(\Omega):=C_c^\infty(\Omega) q\ge 2 Each f\in L^1_{\text{loc}}(\Omega) can be identified with \langle f\rangle\in\mathcal ... 0answers 57 views ### Proof of the Helmholtz-Hodge decomposition Let \Omega\subseteq\mathbb R^3 be open \mathcal D(\Omega):=C_c^\infty(\Omega) Let$$G^2(\Omega):=\left\{\nabla p:p\in L^2_{\text{loc}}(\Omega)\text{ with }\nabla p\in L^2(\Omega)^3\right\}$$... 1answer 46 views ### If p is a distribution, what is the meaning of the claim \nabla p\in L^p(\Omega)^d Let d\in\mathbb N \Omega\subseteq\mathbb R^d be open \mathcal D(\Omega):=C_c^\infty(\Omega) q\ge 1 I've seen the following Lemma (without a proof) in a paper and don't understand how I ... 1answer 33 views ### Dirac functional embedding I got the following statements to show. Let S \neq \emptyset equiped with the discrete topology and let \ell_\infty(S) = \{f: S \to \mathbb C \mid f \text{ bounded}\}. Not \ell_\infty(S) with ... 0answers 21 views ### How to define a variable which is an integral involving cauchy principal value inside? How to define a variable which is an integral involving cauchy principal value inside in any computer programming language? I want to know how to break down the procedure step by step from a ... 2answers 70 views ### A doubt about the vectorial topology on \mathcal{D}(\Omega) We denote with \mathcal{U}_0 the family of all subsets U \in \mathcal{D}(\Omega) convex and balanced such that U \cap \mathcal{D}_K(\Omega) \in \mathcal{T}_K, where \mathcal{T}_K is the ... 1answer 97 views ### How to show for a distribution T and a test function \varphi,~~T'[\varphi]\equiv -T[\varphi']\;? For a generalized function T, we define$$T'[\varphi] ~≡~ −T[φ']~~~~~~\forall φ ∈ \mathcal D(Ω).$$where \mathcal D(\Omega) denotes the test function space. I'm not getting how they ... 2answers 25 views ### Representation of functional on overlapping areas I have given a functional l on C_c^\infty(\mathbb{R}^n). Now let's assume that for any p \in \mathbb{R}^n we have a neighborhood V_p and a 2\pi-periodic C^\infty-function u_p on \mathbb{... 1answer 36 views ### Derivative of principal value distribution 1/x^2 is equal to finite part distribution -1/x^2? Finite part (Partie finie) of the mapping x \mapsto \frac{1}{{{x^2}}} is a regular distribution defined by$$\left\langle {{\text{Pf}}\frac{1}{{{x^2}}},\varphi } \right\rangle = \mathop {\lim }\... 1answer 25 views ### Distributional derivatives for functions that is continuous but nowhere differentiable It is well known that the Brownian motion is an example of functions that is continuous but nowhere differentiable. In addition, its distributional derivative can be interpreted in the way mentioned ... 3answers 97 views ### What is the theory of distribution that makes possible to calculate Fourier transform of the Sine function? I am an Engineering student. Sine wave function is a power signal present from$-\infty$to$+\infty$. I have have read that because of distribution theory,the Fourier transform of the Sine function ... 0answers 30 views ### Is it true that$|∇u(x)|^2\chi_\Omega=|\nabla (u \chi_\Omega)|^2$Let$u\in L^\infty(\Omega)\cap H^1(\Omega)$with$\Omega$open, bounded and regular (as you wish) domain of$\mathbb{R}^N$. Is it true that $$\int_\Omega |\nabla u(x)|^2 \mathrm{d} x= \int_{\mathbb{... 2answers 260 views ### Definition of the convolution with tempered distributions and Schwartz function In the book where I'm studying there is the following exercise. If x \in \mathbb{R}^n, \varphi \in \mathcal{S}(\mathbb{R}^n) and u \in \mathcal{S}'(\mathbb{R}^n) we define (u \ast \varphi)(x)=... 0answers 23 views ### How do we show the inequality for p=\infty? How can we show the inequality for p=\infty ? Since \overline{u} \in W^{1, \infty}(\mathbb{R}^n) we have that \overline{u}' exists and \overline{u}, \overline{u}' are essentially bounded. ... 0answers 20 views ### Show property of convolution Proposition: Let u, v \in L^2(\mathbb{R}^n) then \widehat{u \ast v}=\widehat{u} \cdot \widehat{v}. Proof: We want to show that \mathcal{F}^{-1} (\widehat{u} \cdot \widehat{v})=u \ast v . We ... 0answers 46 views ### Show L^2-convergence Lemma: Let \psi \in C_C^{\infty}(\mathbb{R}^n), \psi \geq 0, \int \psi(x)dx=1, \psi_{\epsilon}(x)= \epsilon^{-n} \psi{\left( \frac{x}{\epsilon}\right)}, \epsilon>0. Let f \in L^2(\mathbb{R}^n) ... 3answers 50 views ### How to solve a differential equation with a distributional free term? I tried to solve this type of differential equation$$y'' + y = \delta + \delta' .$$I tried using the Laplace Transform, but I'm stuck at that$\delta$(Dirac function). The only thing I know is ... 1answer 28 views ### Weak solutions of Navier-Stokes are square integrable distributions? I'm reading Lemarie-Rieusset's book Recent developments in the Navier-Stokes problem and have the following issue: he defines a weak solution to the Navier-Stokes equations on$(0,T)\times\mathbb R^d$... 1answer 70 views ### Find$\sum_{k \geq 1} e^{itk}$in the sense of distribution -$\delta(x-a)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{i(x-a)t}dt$I have to solve$Z(t)=\sum_{k \geq 1} e^{itk}$in the sense of distribution (generalized function), i.e.,$<\sum_{k \geq 1} e^{itk}, \varphi>$, where$\varphi$is a test function. So far, by the ... 1answer 47 views ### Integral of delta function and the constant for fund. solution to laplace's eq When finding the fundamental solution to Laplace's eqn, i.e.$G$such that$\Delta G = \delta$a constant has so be solved for. How I have seen this done is by finding$c$so that$\int_{D(0,\epsilon)...
If $u \in W^{3,p}(\mathbb{R}^{+})$ how can we construct the catoptric extension $\overline{u}$ of $u$ in $\mathbb{R}$ (reflection) such that $\overline{u} \in W^{3,p}(\mathbb{R})$ ? EDIT: By setting \$...