All Questions
Tagged with functional-analysis distribution-theory
1,225 questions
1
vote
1
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85
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Determining when the operator $f \mapsto if'$ is self adjoint with a specific domain.
Considering the Hilbert space for $L^2([0, 1])$ and $z \in \mathbb{C}$, we can define $T_z$ as the closure of $f \mapsto if'$ on the domain $\{f \in C^1([0, 1]): f(1) = zf(0)\}$. I think this domain ...
- 728
8
votes
1
answer
301
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Defintion of distributions why not define with complex conjugate
For a complex valued locally integrable function $f$ on an open set $U \subset \mathbb{R}^{n}$, I saw many sources defined distribution induced by $f$ as $\phi \mapsto \int f\phi \, dx$.
If $\phi$ is ...
- 1,948
2
votes
1
answer
75
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Are probability distributions the same as these distributions in mathematical analysis? (simple question)
Consider this definition of distributions that we see in analysis.
I have been brushing up on my probability and statistics knowledge. I already know that the term "distribution" has ...
- 1,330
2
votes
1
answer
37
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Equivalent characterization of the support of a distribution in $C_0^\infty {'} (\Omega)$.
Before presenting my question in a formal way, I should present some contextualization so my problem is well understood.
Contextualization. Let $\Omega \subset \mathbb R^n$ be a non-empty open set and ...
- 1,217
7
votes
1
answer
189
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Question on distributions and topology
Let $X$ and $Y$ be two topological spaces and $f:X\to Y$ a function. It is well-known that continuity of $f$ implies sequentially continuity, while the reverse is in general only true if $X$ is first ...
- 3,168
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1
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94
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Reed and Simon Chapter IX problem 45 (a): Multiplication by a regular but not smooth function in a negative local Sobolev space
I suppose you could see this as a spiritual follow-up to Multiplication by a regular function in a negative Sobolev space seen as an adjoint, hence the very similar choice of title and the very ...
- 6,596
1
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147
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Understanding a simplification in a proof
While studying the book, ``Generalized function Theory and Technique Birkhauser'', specifically the section on Dirac delta function and Delta sequences, the author uses the sequence
$$s_{m}(x)=\frac{\...
- 1,610
0
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0
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60
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Can the alternating operator be defined for distributions in a compatible way?
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$. The alternating operator $A$ can be defined as follows:
$$Af(x_1, \ldots, x_n) = \frac{1}{n!}\sum_{\sigma \in S_n} \textrm{sgn}(\sigma) f(x_{\sigma(1), \...
- 6,641
1
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0
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35
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About the details of Fundamental solution of wave operator
The following is part of the PDE lecture notes (the first two parts are some definitions and propositions, the key part I want to ask is below the last dividing line). What I want to ask is how the ...
- 769
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0
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44
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Proof of Schwartz kernel theorem
In Melrose's proof of Schwartz kernel theorem here, I have trouble understand the last part of the proof. Basically, suppose $B:H^{-n}(\mathbb R^n_x)\to S(\mathbb R^m_y)$ is a continuous linear ...
- 2,648
1
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2
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66
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Showing the continuity of a complex-valued operator defined on $C_0^\infty(\Omega)$.
Context. Let $\Omega \subset \mathbb R^n$ denote an arbitrary non-empty open set and let $C_0^\infty(\Omega)$ denote the usual space of infinitely differentiable functions that have compact support in ...
- 1,217
1
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1
answer
53
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How to find a distribution $u \in D'(\mathbb{R})$ such that $u(x) = \frac{1}{x}$ for $x > 0$ and $u(x) = 0$ for $x < 0$?
Title:
How to find a distribution $u \in D'(\mathbb{R})$ such that $u(x) = \frac{1}{x}$ for $x > 0$ and $u(x) = 0$ for $x < 0$?
Body:
I'm working on a problem from Introduction to the Theory of ...
- 337
3
votes
2
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98
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Compute $\lim_{n\to\infty}x^{-1}\sin(nx)$ as a distribution in $\mathcal D'（\mathbb R）$
As a distribution on $\mathbb{R}$, $u_n(x) = x^{-1} \sin(nx)$, compute the limit in the distribution space $\mathcal{D}'(\mathbb{R})$ of $\lim_{n \to \infty} u_n$.
I have solved the cases of $u_n(x)=\...
- 769
3
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1
answer
95
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$\lim_{\varepsilon\to0}\frac{2\epsilon}{\pi(x^2+\epsilon^2)}=\delta_0$ in the sense of distribution
Prove that, $\displaystyle\lim_{\varepsilon\to0}\frac{2\epsilon}{\pi(x^2+\epsilon^2)}\overset{\mathcal D'(\mathbb R)}{==}\delta_0$.
$\color{red}{\text{I'm not sure that the exercise is true.}}$ I ...
- 769
2
votes
1
answer
108
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Rudin Functional Analysis Problem 7.22.
I am trying to solve question number $22$ from Chapter 7 titled "Fourier Transforms" of Rudin's Functional Analysis which is about Periodic distributions. I am mostly concered about the one-...
- 25
0
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0
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44
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If $u\in\mathcal D'$ and $\langle u,\phi' \rangle=0$ then there exists $c\in\mathbb C$ such that $\langle u,\phi \rangle= c\int\phi$ [duplicate]
From https://math.stackexchange.com/a/2552946/1084278, we know the following property:
If $u\in\mathcal D'$ and $\langle u,\phi' \rangle=0$ for all $\phi\in\mathcal D$ then there exists $c\in\mathbb ...
- 769
0
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1
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39
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The $\mathbb C$-linear independence of several homogeneous distribution with different degrees
Homogeneous Distribution means a distribution $u\in\mathcal D'(\mathbb R^n)$ satisfying $\langle \lambda^{-d}u,\varphi\rangle=\langle u,\lambda^n\varphi(\lambda x)\rangle$ for every $\lambda>0$. ...
- 769
0
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2
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63
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I don't understand what linearity means in definition of a distribution
Let's take 3 functions:
$f(x) = (x^2 + 1)^2$
$g(x) = \sin(x)$
$z(x) = \delta_0$
Neither of them are linear if using following definition of linearity:
$$f(x+y) = f(x) + f(y)$$
For example, $\delta_0(...
- 149
0
votes
1
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93
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Find all homogeneous distribution $u\in\mathcal D'(\mathbb R^1)$ with degree of $0$ or $1$
Homogeneous Distribution means a distribution $u\in\mathcal D'(\mathbb R^n)$ satisfying $\langle \lambda^{-d}u,\varphi\rangle=\langle u,\lambda^n\varphi(\lambda x)\rangle$ for every $\lambda>0$. ...
- 769
2
votes
1
answer
56
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Fix distribution $u$, find a distribution $\tilde u$ satisfying $x\cdot\tilde u=u$
Prove that, for every distribution $u\in\mathcal D'(\mathbb R)$,there exists a distribution $\tilde u\in \mathcal D'(\mathbb R)$ satisfying $x\cdot\tilde u=u$.
It seems that it can be solved by ...
- 769
0
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0
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63
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The Poincaré-Bertrand Formula for the Heaviside Step Function
The Poincaré-Bertrand formula gives for singular integrals over a smooth curve $\Gamma$ and a function $\phi(t,t_0) = \alpha(t)\beta(t_0)$, where $\alpha\in L^p, \beta\in L^q$ and $q=p/(p-1)$, the ...
- 23
2
votes
2
answers
78
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Proof of the Equivalence of Sequential Continuity and Continuity for a Distribution
Theorem 1.3.2
A linear form $u$ on $C_c^\infty(X)$ is a distribution if and only if $\lim_{j \to \infty} \langle u, \phi_j \rangle = 0$ for every sequence $\phi_j$ which converges to zero in $C_c^\...
- 337
2
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0
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46
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Fourier Transformation of Heaviside Function
Let be $H(x)$ Heaviside function. I want to show that $H$ defines a tempered distribution and compute is Fourier transformation.
The exercise has two steps:
I have alredy proof that $\xi\left(i\hat{H}...
-1
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1
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67
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I don't understand what distribution means
From the Rudin book:
We can therefore assign a " kth derivative " to every f that is
locally integrable: $f^{(k)}$ is the linear functional on $D$ that
sends $\phi$ to $(-1)^k \int f\phi^{(...
- 149
0
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0
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35
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Folland: Extension of a linear map from locally integrable function to distributions
G. Folland in his book "Real Analysis" mentions a general way extending linear operators from functions to distributions, see the highlighted text here. I am sure this must be a theorem in ...
- 25
1
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0
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58
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Pullback by surjective submersion is injective?
Denote by $\mathcal{D}'_X$ the sheaf of distributions on a smooth manifold $X$.
Let $M$ and $N$ be smooth manifolds and $\Phi: M \to N$ a submersion. Then $\Phi$ defines a unique morphism of sheaves $\...
- 4,559
0
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1
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24
views
One proof about variable substitution of generalized functions
Let $\Omega_1$ and $\Omega_2$ be two open subsets of $\mathbb{R}^n$, and assume that the map $\Phi: \Omega_1 \to \Omega_2$ is a diffeomorphism. We establish a relationship that maps a distribution on $...
- 769
1
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0
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46
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Applying the heat semigroup to a tempered distribution
Let $e^{t\Delta}$ be the heat semigroup. Classically, $e^{t\Delta}u$ is the solution to the heat equation at time $t$ with initial condition $u$. In the book Fourier Analysis and Nonlinear Partial ...
- 6,641
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0
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44
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Generalization of delta function identity to product of factors
There is a well-known delta function identity which allows for the expansion of
$$\chi_\epsilon(x)\equiv\left(\frac{1}{\epsilon^2
+x^2}\right)^a,\quad x\in \mathbb{R}^n$$
see for example this Math.SE ...
- 27.2k
1
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0
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43
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Definition of the space $\mathcal{S}'_h(\mathbb{R}^d)$
I am reading the book Fourier Analysis and Nonlinear Partial Differential Equations by Bahouri, Chemin, and Danchin, in which they give the following definition:
We denote by $\mathcal{S}_h'(\mathbb{...
- 6,641
0
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0
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40
views
Square-root of locally integrable functions
Suppose I am given an $L^1_{\text{loc}}(\mathbb{R}^d\times\mathbb{R}^d)$ function $f(x,y)$. It then defines an inner product on the space of test-functions $\mathcal{D}$, via
$$
\mathcal{E}(\phi,\psi)=...
0
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0
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35
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positive eigenfunctions of elliptic PDE
I was wondering the following: Suppose one has a divergence free vector field $u$, and is interested in solutions to the elliptic PDE
$$
\lambda f+u \cdot \nabla f=\Delta f
$$
where $Re(\lambda)>0$...
- 838
1
vote
2
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97
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Distributional Partial derivative .
Question : For any constant real $ K$ , we let $f(x,y)=K-x^2- y^2 $ for $ x^2+y^2 <1$ and $f(x,y)=0$ elsewhere in $\mathbb R^2$
i. compute $D_xf$ and $D_yf$, ( here $D_xf$ means partial ...
2
votes
0
answers
63
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Understanding convolution and distributions
I'm learning about distributions and some important part is its behaviour under the convolution operation. Given a distribution $T\in D'(\mathbb{R}^n)$ and a function $\varphi\in D(\mathbb{R}^n)$, my ...
- 8,013
3
votes
1
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243
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Proof of Dirac delta representation as sum of exponentials
Physicists sometimes use the formula $\delta(x)=\frac1L\sum_{n\in\mathbb Z} e^{2\pi i nx/L}$, where $\delta(x)$ is the Dirac delta "function." I think the rigorous way to interpret this ...
- 6,898
2
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0
answers
52
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Origin of $L_\varepsilon^\nu$ in Lars Hörmander's proof of the kernel theorem in "The Analysis of Linear Partial Differential Operators I"
Context
In the proof of the kernel theorem Hörmander takes open subsets $X_i\subseteq\mathbb{R}^{n_i}$ ($i=1,2$) and relatively compact sets $Y_i\subset X_i$ (that means $\overline{Y}_i$ is compact ...
- 43
1
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1
answer
52
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Convergence of distribution sequence
Let $\{\phi_n\}$ a sequence of mollifiers in $\mathbb{R}$. I want to prove that the distribution $T_{\phi_n}$ associated, i.e., $$T_{\phi_n}(\Phi)=\int_{\mathbb{R}}\phi_n(x)\Phi(x)dx,$$ converges to ...
- 8,013
4
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0
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161
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Another definition for vector-valued distributions
Here, $\Omega\subseteq\mathbb{R}^n$ is an open set, $X$ is a Banach space, and $X^*$ its topological dual. I ended up on a problem for which I need to define a suitable notion for the gradient of a ...
- 905
3
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1
answer
76
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Exact meaning of a bounded set in the space of tempered distributions $\mathcal{S}'$
Let $\mathcal{S}$ be the Schwartz space on some Euclidean space with the family of seminorms given by $\{ \lVert \cdot \rVert_n \}_{n \in \mathbb{N}}$ which gives it the standard Frechet topology. By ...
- 8,078
1
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0
answers
25
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How to prove that a quotient function is constant?
Let $w_1,w_2\in W^{1,p}(\Omega)$ be two functions with $w_1,w_2>0$ and $\dfrac{w_2}{w_1},\dfrac{w_1}{w_2}\in L^{\infty}(\Omega)$, where $\Omega\subset\mathbb{R}^N$ is a bounded domain (i.e. open ...
- 2,090
1
vote
1
answer
39
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Weak derivative of a quotient
Let $w_1,w_2\in W^{1,p}(\Omega)$ with $p\in (1,\infty)$ such that $w_1,w_2>0$ a.e. on $\Omega$ and $\dfrac{w_1}{w_2},\dfrac{w_2}{w_1}\in L^{\infty}(\Omega)$. How can we prove that $\dfrac{w_1}{w_2}$...
- 2,090
0
votes
1
answer
43
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Question about weak derivative of a function
Consider that $w\in W^{1,p}((0,1))$ with $w(x)>0$ for a.a. $x\in (0,1)$. Here $p>1$ is a constant exponent.
Is it necesarily true that $\dfrac{w'}{w}\in L^1(\Omega)$?
My attempt to prove this is ...
- 2,090
0
votes
0
answers
54
views
Are test functions always required when integrating over distributions?
Consider a Gaussian measure $\mu$ over the space of tempered distributions $\mathcal{S}'(\mathbb{R}^n)$. Such a measure is constructed by using cylinder setse of the form
$$C_{d; f_1, \ldots, f_d} = \...
- 6,641
0
votes
1
answer
61
views
Do we need to take PV to define $1/|x|^{3.5}$ as a tempered distribution?
We know that in 1d, $\frac{1}{|x|^{3.5}}$ is not locally integrable around the origin. I learnt from this post that we could apply the following definition to define a tempered distribution:
$$\langle\...
- 341
2
votes
0
answers
64
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The solution of the heat equation for data in $\mathcal{S}'$.
Compute the fundamental solution of the initial value problem for the heat equation,
$$ \frac{\partial G}{\partial t} - \Delta G = 0, \quad (x,t) \in \mathbb{R}^d \times (0,\infty), $$
with the ...
- 1,682
1
vote
1
answer
31
views
Norm on the space of rapidly decreasing and continuous functions
In P.Malliavin’s book "Integration and probability" a continuous function $f$ defined on $\mathbb{R}^n$ is said to be of rapidly decrease if for all integer $m$ we have that the mapping $(1+\...
- 1,641
4
votes
1
answer
86
views
Is it true that $\widehat{(\delta_{x_{0}}\otimes T)} = \hat{\delta}_{x_{0}}\otimes \hat{T}$?
For a fixed $x_{0} \in \mathbb{R}$ consider the Dirac delta distribution $\delta_{x_{0}}$. Its Fourier transform is given by $\hat{\delta}_{x_{0}}(p) = e^{-px_{0}}$, in the sense that $\hat{\delta}_{...
- 987
2
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0
answers
86
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Understanding spaces of negative regularity
Let $C^k(\mathbb{R}^n$) be the space of functions with $k$ continuous derivatives, and $H^s(\mathbb{R}^n)$ the Sobolev space $W^{2,s}$. Their dual spaces are commonly denoted as $C^{-k}$ or $H^{-s}$. ...
- 6,641
1
vote
0
answers
45
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Possible to define an inner product on tempered distributions of compact support?
I am trying to understand why, in the context of reproducing kernel Hilbert spaces, there seems to always be a square-energy restriction on bandlimited functions in the Paley-Wiener space. (I get why ...
1
vote
0
answers
47
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Density of Schwartz distributions in the space of distribution
Let $S(R^3)$ and $D(R^3 )$ be the space of Schwartz function and test function respectively, $S'(R^3)$ and $D'(R^3)$ be their duals.
I want to understand what "$S'(R^3)\subset D'(R^3)$ and the ...
- 191