Questions tagged [distribution-theory]
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).
3,511
questions
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22
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Making a Fourier Transform converge
Consider for example the following function
\begin{equation}
f(x)=(e^x+1)^{ik_1}\,.
\end{equation}
For $k_1 \in \mathbb{R}$, this is clearly not absolutely integrable and thus its Fourier transform ...
0
votes
1
answer
51
views
How do we derive the composed Dirac Delta function?
Suppose we have to compute $∂_{x}\theta(y-x)$, where $θ$ is the Heaviside step function, how do I find the result?
Intuitively I would just use the chain rule
$\partial_{x}\theta(y-x)=-\delta(y-x)$
...
- 173
0
votes
2
answers
75
views
Counterexample concerning Rudin's definition of open sets in $\mathscr D(\Omega)$
In Rudin's Functional Analysis, in definition 6.3, he defines a family $\beta$ subsets of $\mathscr D(\Omega)$ as follows.
$\beta$ is the collection of all convex balanced sets $W \subset \mathscr D(\...
- 6,422
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votes
0
answers
25
views
Orders of terms of converging sequence of distributions
Let $\Lambda_i$ be a converging sequence of distribution in $D (\Omega)$ for some $\Omega$ open set in $\mathbb{R}^n$.
I want to prove that the order of all $\Lambda_i$ is bounded.
Since the sequence ...
- 73
2
votes
1
answer
58
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Bounded sequence in $H^s$ that has no convergent subsequence.
I'm trying to solve the following problem from Folland's book (exercise 9.36).
Suppose that $0 \neq \phi \in C_c^{\infty}(\mathbb{R}^n)$ and $\{a_j\}$ is a sequence in $\mathbb{R}^n$ with $|a_j|\...
- 89
1
vote
0
answers
37
views
Laplacian of 1/r^n in the distributional sense
Let $\Omega = \mathbb{R}^3\setminus\{0\}$. Consider the function
$$
f_n \colon \Omega \to \mathbb{R},\quad \vec{x} \mapsto \frac{1}{\|\vec{x}\|^n}
$$
with $n \in \mathbb{Z}^{+}$.
I want to calculate ...
- 43
1
vote
0
answers
32
views
Who is "the set of weakly differentiable functions"?
My question is pretty straightfoward: I would like to be able to think of "the set of weakly differentiable functions". Recall that:
a function $f:I=(a,b) \rightarrow \mathbb{R}$ is weakly ...
- 643
0
votes
1
answer
33
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Derivative of a distribution, including an integral of abs(x)
I'm currently reading a book on finite element analysis and I lack knowledge on higher level of mathematics. My question is regarding the derivative of a distribution. I'm stuck on an example ...
- 1
0
votes
1
answer
54
views
What is the limit for $x$ approaching zero of the dirac delta distribution $\delta(x)$?
I am trying to prove that
$$
\lim_{y\rightarrow 0} \delta(y-x) = \delta(x)
.$$
To justify this problem, this comes from the orthogonality of the position eigenstates in quantum mechanics. Indeed, we ...
- 3
1
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0
answers
41
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Is the convolution of a tempered distribution and a Schwartz function also a function?
Let $T \in \mathscr{S}'(\mathbb{R}^n)$ and $f \in \mathscr{S}(\mathbb{R}^n)$. We may define their convolution as
$$(T * f)(\varphi) = T(\tilde{f}*\varphi)$$
where $\tilde{f}(x) = f(-x)$. The above ...
- 4,311
1
vote
1
answer
37
views
Evaluating $\int_{p} f(p) \delta(p+a) \Theta(p)= f(-a) \Theta(-a)$ for Heaviside theta function
I want to evaluate the following integral
For given arbitrary function $f$ and Dirac Delta function $\delta$ with Heaviside Theta function $\Theta$, what is the form of
\begin{align}
\int_{p} f(p) \...
- 6,326
0
votes
1
answer
86
views
How can I get the properties of a function, which has an integral equal to zero?
Let $\varrho \in L^\infty_{loc}$, where the set of discontinuity points is a Lebesgue null set, and $\varrho * \varphi$ is a polynomial with maximum degree of $m$ $\forall \varphi \in C_c^\infty$.
Now ...
- 33
3
votes
1
answer
74
views
Interpolation inequality - what does it mean?
Suppose $f_{k} \to f$ strongly in $L^{2}(\mathbb{R}^{n})$. Let $2 \le 2q < \eta$ where $\eta = \infty$ if $n \le 2$ and $\eta = 2n/(n-2)$ if $n \ge 3$. During a proof, my professor wrote that $\|f_{...
- 695
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0
answers
29
views
Fundamental solution of a differential operator in a Schwartz space
It is given $Lu = u''' -3u'' + 2u'$ in Schwartz space, i.o. $\mathcal{S}(0,+\infty)$ with boundary conditions $u(0)=0$
My attempt is:
Fundamental solutions depends on 2 variables so $Lu=\delta_y(x)$ ...
- 121
1
vote
2
answers
63
views
What technique is to use for differentiation regarding $\delta$-functions?
Interesting differentiation regarding $\delta$-functions. Let we define a function $h(x)=e^{-ax}H(x)+e^{ax}H(-x)$ and we want to find the $n$-th derivative.
\begin{align}
h(x)&=e^{-ax}H(x)+e^{ax}H(...
- 783
0
votes
0
answers
43
views
Derivative of Dirac delta
To represent formally the part of an algorithm
if $r \in [0,p (\mathbf{v})]$, set $y = A(\mathbf{v})$; else set $y = B(\mathbf{v})$,
where $\mathbf{v}$ is a vector of parameters and $r$ is a uniform ...
- 668
2
votes
0
answers
39
views
L. Hörmander: More general Fourier-Laplace transforms
At the beginning of Chapter 7.4 of Volume 1, the concept of the Fourier-Laplace transform on $\mathscr E'(\mathbb R^d)$ is generalized:
"For distributions $u$ with compact support we have defined ...
- 21
1
vote
1
answer
50
views
Weak derivative for $\mathrm{L}^p$-functions
Suppose we have a function $f \in \mathrm{L}^p(\Omega)$ for an open set $\Omega \subset \mathbb{R}^n$, such that the distributional derivatives for all multi-indices $\alpha$ up to degree $m$ are ...
- 13
2
votes
1
answer
37
views
Distribution of the combination of square and the product of Gaussian random variables [closed]
What is the distribution of the following expression?
$$
\sum_{i=1}^n a_i x_i^2 + \sum_{i \neq j; i,j=1}^n b_{ij} x_i x_j
$$
where $x_i$ for $i$ in the range from 1 to $n$ are i.i.d. samples from $\...
- 119
1
vote
0
answers
40
views
Multiple change of variables in integral to difference between variables
When changing variables in multiple integrals one needs to find the Jacobian matrix of the transformation
$$\int dx_1\cdots dx_n f(x_1,...,x_n)=\int dy_1\cdots dy_n f(y_1,...,y_n)\left|\frac{\partial(...
- 29
0
votes
1
answer
70
views
Derivation in the sense of distributions
Let $f_n(x)$ be a continuous sequence functions.
Assume that the series $\sum_{n\in N} f_n(x)$ converges simply.
Put $f(x)=\sum^\infty_{n=0} f_n(x)$.
My question if the function $f$ is continuous.
Is ...
- 93
4
votes
0
answers
51
views
Changing order of integration in relation to the Fourier Transform
In a lot of my engineering textbooks the order of integration on improper integrals is mindlessly swapped in proofs regarding properties of the Fourier Transform. This has always bothered me: when I ...
- 41
1
vote
2
answers
55
views
Integration of Dirac delta
According to Mathematica,
$$\int_{0}^{b}f(x)\delta(x-a)\,dx\,=\,f(a)\,(2\Theta(b)-1)\,\Theta(a-b\Theta(-b))\,\Theta(-a+b\Theta(b)).$$
for $a,b\in\mathbb{R}$. Is there a quick way to see how this ...
- 419
4
votes
1
answer
54
views
Taylor series in distribution sense?
If $f$ is integrable, $\partial_x f$ makes sense in the distributional sense as $(\partial_x f,\phi)=(f,\partial_x \phi)$
Question. Does a sort of Taylor series make sense distributionally?
that is, ...
- 3,220
0
votes
1
answer
71
views
Singular differential equation
The differential equation
$$\frac{dy}{dx}\,=\,\frac{A}{x}$$
has the general solution:
$$y\,=\,\begin{cases}
\begin{array}{c}
A\log(x)+B\\
A\log(-x)+C
\end{array} & \begin{array}{c}
x>0\\
x<0,...
- 419
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votes
0
answers
44
views
How to prove$|\cdot|^{s}g\in\mathcal{S}'$? a question from Bahouri-Chemin-Danchin book "fourier analysis and nonlinear pde"
In page 27 and 28 of book "fourier analysis and nonlinear partial differential equations", proposition 1.36 the authors Bahouri-Chemin-Danchin give a proof of this proposition, but I really ...
2
votes
1
answer
77
views
Distributional Laplacian of Logarithmic function.
Here is the exercise:
Compute the distributional Laplacian $\left(\text{in }\mathbb{R}^2\right)$ of $d(x,y)=\ln\left(\|(x,y)\|\right)=\ln\left(\sqrt{x^2+y^2}\right)$. Relate your answer to $\delta$ ...
- 35
4
votes
1
answer
77
views
Dividing a tempered distribution by a polynomial
Let $p=p(x_1,...,x_N)$ be a non-zero polynomial in $N$ variables (real coefficients). Let $\mathscr{S}$ be the Schwartz space on $\mathbb{R}^N$ and let $\mathscr{S}'$ be its topological dual (i.e. the ...
- 3,537
0
votes
0
answers
94
views
Sum in the distributional sense
Put
$$ S_n(\xi, x)=\sum_{k=0}^{+\infty} \frac{L_k^n(x)}{\xi-k}, x \in C, \xi \in C \backslash Z_{+} $$
Now, to compute the sum $S_n(\xi, x)$ we begin by writing it in terms of the Laguerre polynomial ...
- 93
3
votes
2
answers
145
views
Generalization of Dirac delta identity
The Dirac delta distribution obeys the following identity in $\mathbb{R}$
$$\lim_{\epsilon\to 0}\dfrac{1}{\pi}\dfrac{\epsilon}{\epsilon^2+x^2}=\delta(x)\tag1.$$
I know how to prove this using the ...
- 25.6k
1
vote
1
answer
127
views
From Trevés - Finite order distributions in terms of Radon measures
Trevés, Theorem 24.4, Topological Vector Spaces, Distributions and Kernels, Dover, p. 259.
If $T$ is a distribution of finite order $\leq m$ in $\Omega$ with support $S \subset \Omega$, then given any ...
- 245
2
votes
1
answer
158
views
$d$-dimensional Fourier transform of $\cos|x|$
What is the $d$-dimensional Fourier transform of $\cos|x|$? More specifically, what does
$$
\int_{\mathbb{R}^d}\text{d}^dx\cos|x|e^{-i\vec{k}\cdot\vec{x}}\ ,
$$
where $|x|=\sqrt{x_1^2+\dots+x_d^2}$,
...
- 81
0
votes
1
answer
39
views
Calculate the limit of $\langle \partial_x f,\rho_\varepsilon \ast \chi_B \rangle $
Let $f(x_1,x_2)=x_2^2\chi_E$ where $E=\{(x_1,x_2)\in \mathbb{R}^2:x_1\geq 0 \}$ and let $B$ be the unit ball in $\mathbb{R}^2$. Since $f$ is locally integrable, it can be interpreted as a distribution ...
- 3,537
2
votes
1
answer
70
views
Regarding a non-standard definition of tempered distributions in Schuller's lecture
Is the below definition of tempered distributions correct?
$\newcommand{\rr}{\mathbb{R}}
\newcommand{\cc}{\mathbb{C}}
\newcommand{\nn}{\mathbb{N}_0}
\newcommand{\schwartz}{\mathcal{S}}
\newcommand{\...
- 140
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votes
0
answers
55
views
Complex conjugate of a delta function
How should we regard to the complex conjugate of a Dirac delta function? I mean, is the observation that
$\left(\delta(x)\right)^{\ast}\neq\delta(x),\quad\left(\Theta(x)\right)^{\ast}\neq\Theta(x)$
...
- 419
0
votes
0
answers
46
views
Why characteristic functions of regular domains are not in $W^{1,p}$?
Let $E\subset\mathbb{R}^n$ be a bounded domain with $\partial E\in C^1$.
I don't know how to prove that its characteristic function $\chi_E$ is NOT in $W^{1,p}(\mathbb{R^n})$.
It's obvious that $\...
0
votes
1
answer
31
views
Absolutely continuous function and distribution theory
Let $f$ and $g$ be a locally integrable function on $[0,1]$. For any $\phi\in C^1[0,1]$, $f$ and $g$ satisfies
$\displaystyle \int_{0}^{1}\phi'(t)f(t)dt+\int_{0}^{1}\phi(t)g(t)dt = \phi(1)f(1)-\phi(0)...
- 1,381
0
votes
3
answers
46
views
$(f\delta')(\phi)=-(f\phi)'(0)$? Or $-(f\phi')(0)$?
This might be a stupid question but I can't find the answer elsewhere.
Put $\delta(\phi)=\phi(0)$ for $\phi\in\mathcal{D}(\mathbb{R})$, the test function space on $\mathbb{R}$, and $f\in C^\infty(\...
- 33
1
vote
1
answer
71
views
Fourier integral representation of dirac delta function
Is the following proposition true?
Proposition. For any $a,k\in \mathbb{R}$,
\begin{equation}
\int_a^{\infty} dx e^{ikx} = 2\pi \delta(k).
\end{equation}
(End)
I think it is true based on the ...
- 290
1
vote
0
answers
30
views
defintion of weak solution
i found that a weak solution of $-\Delta u=f $ where f $\in L^{2} (\Omega)$ is a solution u $\in H_{0}^{1}(\Omega)$ and it is solution in sense of distribution so what does it mean that that u is ...
- 11
2
votes
1
answer
97
views
Confused About Rudin's Opening Statements on his Chapter on Distributions
I have a few doubts regarding Rudin's opening paragraphs in Chapter 6 of Functional Analysis. The passage (with small edits) reads:
A complex function $f$ is said to be locally integrable if $f$ is ...
- 4,708
0
votes
1
answer
34
views
Upper bound of sum of random variables
Suppose that we have random variables $X_1$, $X_2$ each drawn independently from Irwin Hall Distribution with same mean of 0 and different variances.
If I have B1 and B2, which are high probability ...
- 3
1
vote
0
answers
80
views
Does all the well defined product of Hyperfunction satisfy the product rule
I read in the first two chapters of Urs Graf's "Introduction to Hyperfunctions and Their Integral Transforms". On the page 114 of the book, The book well defined the product of ...
- 23
0
votes
1
answer
43
views
Is a function with good properties defined in a closed interval a generalized function?
I'm not a math student. I just want to touch distribution theory briefly. My question is, is a function with good properties defined in a closed interval a generalized function?
For example, $f(x)=x$ ...
- 23
6
votes
2
answers
150
views
Confused about Georgiev's definition of $C^{\infty}_c$
I'm beginning to study Distributions, and I've encountered the following definition in Georgiev's Theory of Distributions:
Such definition implies that $C^{\infty}_c$ (with the opology given by the ...
- 4,708
1
vote
0
answers
71
views
What is the easiest way to define the canonical $LF$-topology?
There are many ways to define the canonical $LF$-topology on $C^{\infty}_c$, as explained here.
Understanding that none of them is particularly easy, which method for defining the topology is the ...
- 4,708
2
votes
1
answer
47
views
Distribution against non-zero function
I often see equation following:
$$\int_{-\infty}^{\infty} \delta(x) dx=1 =\int_{-\infty}^{\infty} 1 \cdot \delta(x) dx $$
But, as i know, The distribution isn't possible for any function; it is only ...
- 59
0
votes
0
answers
31
views
condition for the Dirac delta function to be in Hilbert space
I have a problem to show $$ \delta\in\mathcal{H}^s(\mathbb{R}^n) \iff s<-\dfrac{n}{2}. $$
where $ \mathcal{H}^s(\mathbb{R}^n)= \{f\in\mathcal{S}'(\mathbb{R}^n): \Vert(1+\vert\xi\vert^2)^{s/2} \...
- 7
0
votes
0
answers
35
views
What is a Dirac bi-density?
In https://arxiv.org/abs/gr-qc/9210011 on p. 22 the author encounters the Dirac bi-density $\delta(x,x')$ in certain Poisson bracket relations in Hamiltonian mechanics.
He says that $\delta(x,x')$ is ...
- 473
1
vote
2
answers
63
views
Product of dirac delta function without integral
I am learning Distribution theory.
I see that $\int_{-\infty}^{\infty} f(x) \delta(x) \,dx = f(0)$
The question is, at x=0, is $f(x) \delta(x) = f(0)$, even without integration?
Because at one place, ...
- 103