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).
3,645
questions
2
votes
1
answer
360
views
What would be to show $(1+|\xi|^2)^{s/2}\hat{u}\in \mathcal{S}^{'}(\mathbb R^n)$ is a function?
If $s\in\mathbb R$ define the sobolev space $H^s(\mathbb R^n)$ as $$H^s(\mathbb R^n):=\{u\in \mathcal{S}^{'}(\mathbb R^n): (1+|\xi|^2)^{s/2}\hat{u}\in L^2(\mathbb R^n)\}.$$ I have a doubt concearning ...
- 9,655
2
votes
1
answer
1k
views
Distribution with finite support. [duplicate]
If $f \in\mathcal D′(\Bbb R^n)$, is a distribution and support of this
distribution is a set of finite points in $\Bbb R^n$. Can anyone tell
me that what will be the general form of this ...
- 2,068
2
votes
2
answers
76
views
Convergence of a sequence in $L^1(\mathbb{R}^3)$
All function spaces are over $\mathbb{R}^3$.
Let $u_n \in C^\infty_0$, $u_n\rightarrow u$ in $L^1$. Let $v\in L^1_\text{loc}$ be such that $uv \in L^1$. Does $u_n v \rightarrow uv$ in $L^1$?
What ...
- 1,539
2
votes
1
answer
3k
views
Proving the mean value property of harmonic functions using distributions?
A professor I talked to showed me a proof of the mean value property. (He actually showed it for functions solving the heat equation instead of Laplace's equation, but it seems like the argument is ...
- 2,020
2
votes
1
answer
557
views
What does it mean to say an integral exists 'in the distributional sense'?
What exactly does it mean to say that an integral exists 'just in the distributional sense'? For example, the Fourier transform of $x^2 e^{-\lambda x}$ or of $H(R-|x|)$ where $R > 0$ and $H$ is the ...
- 18.1k
2
votes
1
answer
603
views
Proving that the bi-laplacian of a radial basis function is the dirac delta
According to equation (2.14) of the paper "The Uniform Convergence of Thin Plate Spline Interpolation in Two Dimensions" a radial basis function $\phi(\parallel x \parallel)$ has the property
$$
\...
- 1,241
2
votes
1
answer
76
views
Differential equation with Fourier transform
It is given the following differential equation:
$$-\frac{d^2 x}{dt^2} + \frac{dx}{dt} = \theta(t) e^{-\varepsilon t}$$
where $\theta(t)$ is the Heaviside function (which equals $0$ for $t<0$ and ...
- 97
2
votes
1
answer
71
views
What Hardy's inequality is this?
During a seminar the other day, the lecturer mentioned a Hardy's inequality $|x|^{-2} \le K(1-\Delta)$, where $K$ is a constant and $\Delta$ is the Laplacian in $\mathbb{R}^{3}$. I searched for Hardy'...
- 869
2
votes
1
answer
111
views
Why is the solution of the Klein-Gordon PDE a distribution?
I've also posted this question on physics SE in case it is more appropiate there.
Consider the Klein-Gordon equation:
$$(\square + m^2)\phi = (\partial_t^2 - \Delta + m^2)\phi = 0 \tag{1}$$
The ...
- 5,965
2
votes
1
answer
88
views
Are all distributions of the form $ \varphi \mapsto \int \varphi^{(n)} d\mu $?
Distributions are defined as the set of linear continuos operators over the set of test functions $ \mathcal{D}(\mathbb{R}) = C^{\infty}_c(\mathbb{R}) $, i.e.
$ \mathcal{D}(\mathbb{R})^* $.
My ...
- 769
2
votes
1
answer
35
views
derivative of the functions in Hadamard's lemma
I think Corollary 1.10 of this PDF page 12 is Hadamard's lemma
here $\partial_jf$ is the partial derivative $\frac{\partial f}{\partial x_j}$ and $\alpha$ is a multi-index. $\partial^\alpha$ is ...
- 2,345
2
votes
1
answer
62
views
The derivative of the Heaviside step function in $\mathbb{R}^n$
Here is a problem I have:
Consider the Heaviside function
$$H(x)=H(x_1)H(x_2)\cdots H(x_n), ~~~x\in\mathbb{R}^n$$
and prove that
$$\dfrac{\partial^n H(x)}{\partial x_1\partial x_2\cdots\partial x_n}=\...
- 29
2
votes
1
answer
41
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 $\...
- 219
2
votes
1
answer
55
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 ...
- 69
2
votes
1
answer
98
views
Translation operator on tempered distributions converges to derivative operator
Let $\mathscr{S}(\mathbb{R})$ be the Schwartz space over $\mathbb{R}$ and $\mathscr{S}'(\mathbb{R})$ the space of tempered distributions. Define an operator $U_a$ on the space of tempered ...
- 5,965
2
votes
1
answer
48
views
What is the Laplace transform of the distribution $\delta \left(\cos \left(\frac{1}{x}\right)\right)$?
Trying to find numerocity of the roots of the function $\cos(1/x)$ I stumbled at the following problem: what is the Laplace transform of the distribution $\delta \left(\cos \left(\frac{1}{x}\right)\...
- 9,119
2
votes
1
answer
594
views
Integrating the dirac delta function multiplied with another function
I'm trying to evaluate
$$
\int_{-\infty}^{\infty}dxf(x)\delta(g(x))
$$
Where $\delta$ is the dirac delta function, and $g(x)$ has zeros at $\{x=x_n, n = 1,...,N\}$. Here're some of my steps:
$$
\int_{-...
- 919
2
votes
2
answers
57
views
$ \frac{1}{2\pi } \int dk e^{ik x } = \delta(x)$ in the distributional sense
It is often said or heard. The right hand side is well understood as a distribution or linear functional.
How to make sense of the left hand side as a distribution?
- 1,005
2
votes
1
answer
51
views
Prove that $\eta^\epsilon(x)$ converges to Dirac delta
I would like to prove that
$$\eta^\epsilon(x):=\begin{cases}\displaystyle\frac{\epsilon-|x|}{\epsilon^2}&\text{, if }|x|\le \epsilon\\0&\text{, if }|x|\ge \epsilon\end{cases}$$
converges to ...
- 7,851
2
votes
1
answer
330
views
Fourier transformation of tempered distribution $p.v. \frac{1}{x}$
Prove the fourier transformation of $p.v. \frac{1}{x}\in \mathcal{S}'(\Bbb{R})$ is $-i\pi \text{sgn}(y)$.
This is a classical result that shows in many textbook,my question is some detail question ...
- 4,786
2
votes
2
answers
192
views
What is the "Main Value of an Integral" (denoted as $\textbf{VP}\int_{-\infty}^{\infty} \frac{\phi(x)}{x}dx$ for $f(x) = \frac{\phi(x)}{x}$)?
The definition given to us is as follows (for $\text{supp }\phi \subset (-R, R)$ ):
$$\textbf{VP }\int_{-\infty}^{\infty} \frac{\phi(x)}{x}dx = \lim_{\varepsilon \rightarrow 0^+} \: (\int_{-R}^{-\...
- 555
2
votes
1
answer
35
views
$C^{\infty}$ reconciliation of two functions
Let $\Omega$ be an open subset of $\mathbb{R}^d$, $K$ a compact subset of $\Omega$ and $\phi \in C^\infty(\Omega)$. Given $\epsilon > 0$ and denoting $K_\epsilon = \{ x \in K : d(x,K^c) \geq \...
- 2,007
2
votes
1
answer
101
views
Weak convergence and Laplace's equation
Suppose $\{u_i\}$ is a sequence of distributions that satisfies Laplace's equation
$$\Delta u_i = 0, \forall i$$
in the weak sense. If $\{u_i\} \to u$ in the sense of distribution, show that $u$ also ...
- 583
2
votes
1
answer
396
views
What are distributions really?
If they are considered to be generalisation of function, as for examples real numbers are for rationals, then it must be possible too reinterpret any “classical” function $f(x)$ in terms of a ...
- 121
2
votes
2
answers
147
views
Show that the finite part of $\frac{1}{x_+}$ is a distribution
I would like to show that the finite part of $\frac{1}{x_+}=\mathbb{1}_{[0,+\infty[}\frac{1}{x}$, noted Fp$\left(\frac{1}{x_+}\right)$ and given by $$\left\langle\text{Fp}\left(\frac{1}{x_+}\right),\...
- 613
2
votes
1
answer
81
views
If $u \in C$, $|\int u \phi_{ij}| \leq 2 \|h\|_p \|\phi\|_{p'} \forall \phi \in C_c^\infty$, then $\|u_{ij}\|_p \leq 2\|h\|_p$?
Let $\Omega$ be a domain in $\mathbb R^N$, $u \in C(\Omega)$ and $h \in L^p(\Omega)$. Suppose that
$$
\left|\int_\Omega u \varphi_{ij} \ dx \right| \leq 2\|h\|_p \|\varphi\|_{p'} \quad \forall \...
- 3,046
2
votes
2
answers
544
views
Dirac Delta Function "removing integral"
I am trying to proof the following identity:
\begin{align}
\delta(ax) = \frac{1}{|a|}\delta(x).
\end{align}
where $\delta$ represents the dirac delta function.
I found that
\begin{align}
\int_{-\...
- 21
2
votes
2
answers
137
views
About the definition of $\phi_k\xrightarrow{D}\phi$ (distribution theory)
Let $\phi\in \mathcal{D}(\Omega):=C_c^{\infty}(\Omega)$ and $(\phi_k)_k$ a sequence of functions in $\mathcal{D}(\Omega)$
We define $\varphi_k \rightarrow \varphi$ in $\mathcal{D}(\Omega)$ as
(1) $\...
- 4,206
2
votes
1
answer
126
views
Proving a sequence of locally integrable functions goes to Dirac delta
Attempt: I have done this type of questions before too and usually a substitution $x =\frac{x}{m}$ does the job. But it does not work in the case. How should I proceed?
Any hints will be appreciated!
- 620
2
votes
3
answers
534
views
Limit in distributions of $\frac{\sin(tx)}{x}$
How can I find the limit of $\frac{\sin(tx)}{x}$ as $t \to \infty$ in $D'$ ? I understand that i need to see the $\lim_{t \to \infty}{\int_{\infty}^{\infty}{\frac{\sin(tx)\phi(x)}{x}dx}}$ for every ...
2
votes
1
answer
612
views
What is the asymptotic form of the spherical Bessel function $j_{n}(x)$ when $n \to \infty$?
I am trying to find the asymptotics of the spherical Bessel function $j_n(x)$
when $n\to \infty$.
I was able to find something like
$$ j_{n} ( x) \sim \sqrt{\frac{\pi}{2n}} \delta \left(x - n\...
- 652
2
votes
1
answer
272
views
Criteria for a sequence of funcions converge to Dirac Delta
I have been stuck with the following problem:
Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of integrable functions in $\mathbb{R}^N$ that satisfies:
$$\int_{\mathbb{R}^N}f_n(x)dx = 1 \quad \forall n \...
2
votes
1
answer
143
views
Gelfand triples/Rigged Hilbert space notation
Usually the rigged Hilbert space is denoted by $\mathcal{S} \subset L^2 \subset \mathcal{S}'$, where $L^2$ is a Hilbert space (square integrable functions), $\mathcal{S}$ is the Schwartz space and $\...
- 357
2
votes
1
answer
58
views
Test functions and distributions
Let $a \in C^\infty(\Bbb R)$. Show that if $g \in D'
(\Bbb R)$ satisfies $g' − ag = 0$, then $g \in C^\infty(\Bbb R)$,
i.e. $g$ is a regular distribution associated with $C^\infty$ function.
I have ...
2
votes
1
answer
82
views
Why do we have $\int_{t_0}^t \delta(t-t') c(t')=\frac{1}{2} c(t)$?
I am reading a physics paper in which they claim that:
$$\int_{t_0}^t \delta(t-t') c(t')=\frac{1}{2} c(t).$$
They just say about this equation:
"The result will always hold when $\int_{-\infty}^...
- 689
2
votes
1
answer
1k
views
Find weak form of linear transport equation
I am stuck on the following problem that says:
a) Find a weak formulation for the partial differential equation $${\partial u\over\partial t\ }+ c{\partial u \over \partial x\ }=0$$
b) Show ...
- 163
2
votes
3
answers
3k
views
If X is a non-negative continuous random variable, show that $E[X]=\int_0^\infty (1-F(x)) dx$
There is a hint to solve this by using integration by parts. So I have $$ u = x \space \mbox{then} \space u'=1$$
$$v' = F(x) -1 \space \mbox{then} \space = ? $$
QUESTION:
What is $v'$ ?
What is ...
- 481
2
votes
2
answers
67
views
Why $\frac{\varepsilon}{x^2+\varepsilon^2}$ converges in the sense of distributions to a constant times the Dirac delta
The integral of $f_\varepsilon(x)=\frac\varepsilon{x^2+\varepsilon^2}$ is the tan inverse, which is well behaved anywhere on $\mathbb{R}$, and so $f_\varepsilon$ is in $L^1_\text{loc}(\mathbb{R})$. ...
- 846
2
votes
1
answer
524
views
Extension of a distribution
Let $\Omega \subseteq \mathbb R^n$ be an open convex set and $T \in D'(\Omega)$. Is it possible to extend $T$ to a distribution in $D'(\mathbb R^n)$?
I've been looking in some texts (Rudin and this ...
2
votes
1
answer
626
views
CDF for exponential family
Might be a stupid question though...
we know for Exponential Family, we have the density is
$$f(x|\theta)=h(x)\exp[\eta(\theta)\cdot T(x)-A(\theta)]$$
Is there a general form for the CDF function? I ...
- 2,088
2
votes
1
answer
316
views
In what sense are distributions more singular than functions?
Chapter 9 in Folland's Real Analysis starts with the following sentence:
"At least as far back as Heaviside in the $1890$s, engineers and physicists have found it convenient to consider ...
- 1,241
2
votes
1
answer
76
views
Proving existence of a Brownian functional
Let $W$ be a standard Brownian Motion indexed by $\mathbb{R}$, i.e., $W\restriction_{\mathopen{[}0,\infty\mathclose{[}}$ is a standard Brownian Motion, and $W\restriction_{\mathopen{]}-\infty,0\...
2
votes
2
answers
6k
views
Formal derivation of the Fourier transform of Dirac delta using a distribution
The Fourier transform of Dirac delta is often naively calculated by considering Delta function as a function that makes sense within an integral and by using its fundamental property:
$$
\int_{-\...
- 405
2
votes
2
answers
180
views
Is there a sense in which this limit is zero?
This question arose from this answer of mine on DSP.SE. Basically, I need to check if this is true:
$$\lim_{M\to \infty} j\frac{\cos[\omega(M+1/2)]}{2\sin(\omega/2)}\stackrel{?}{=}0$$
where $j$ is ...
- 798
2
votes
2
answers
476
views
Prove that , $e^{f(x)}-1$ is a schwartz class function when $f$ is a Schwartz function.
Suppose $f$ is a Schwartz function, prove that $e^{f(x)}-1$ is also a Schwartz function.
In order to prove the result, we have to show that $|(e^{f(x)}-1)^{(n)}|\leq \frac{C_{m,n}}{|x|^{m}}$ for any ...
- 2,017
2
votes
1
answer
277
views
Laplace(-Beltrami) operator in the sense of distributions
Let us set ourselves in $\mathbb R^d$ and let $\Omega$ be an open set. Let $P$ be a differential operator with coefficients that admits a fundamental solution $E$ which is $C^\infty$ outside of $0$. ...
- 2,060
2
votes
1
answer
414
views
Multiplication of two distributions whose singular supports are disjoint
The Wikipedia page on distributions says that multiplication of two distributions whose singular supports are disjoint is easy to define, but the page doesn't actually define it. I'm not sure how to ...
- 681
2
votes
1
answer
94
views
Showing that $f(z)$ is analytic.
Let $x\in \Bbb R^n, y\in\Bbb R$, and $z\in\Bbb C$. For a fixed $\lambda\in\Bbb R^n$ and a smooth, compactly supported function $\varphi\in\mathcal D(\Bbb R^{n+1})$, I want to show that
$$
f(z):=\int_{\...
- 15.1k
2
votes
1
answer
123
views
Fourier Transform of $\sin(\ln(t))u(t) $
Can someone help me with the example of Fourier transform
$$\mathcal{F}[\sin(ln(t))u(t)](\omega)\ = \ \ ?$$
I know that there is a formula in the Table of Fourier Transform Pairs but it is only for $...
- 43
2
votes
1
answer
188
views
General theory of divergent, highly oscillatory integrals
I'm interested in evaluation of integrals of the form $$\int e^{i S(x)}O(x) dx,$$
where $S(x)$ is a polynomial of degree higher than one and $O(x)$ is a polynomial or more generally, polynomialy ...
- 3,070