Use this tag for questions about Schwartz distributions, also known as Generalised 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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3
votes
1answer
573 views

Prove that $f$ does not have a weak derivative

Consider a function $f:\mathbb{R} \rightarrow [0,1 ]$ defined by: $\begin{equation*} f(x)=\left\{ \begin{array}{rl}0 & \text{if } x\leq 0,\\ 1 & \text{if } x\geq 1, \\ 1/2 & \text{if } ...
4
votes
1answer
114 views

Distinction between “measure differential equations” and “differential equations in distributions”?

Is there a universally recognized term for ODEs considered in the sense of distributions used to describe impulsive/discontinuous processes? I noticed that some authors call such ODEs "measure ...
1
vote
1answer
462 views

differential equation with distributions

I'm stuggeling with this differential equation: $T'+T=0$ Where $T$ is distribution. I found solutions in form: $\sum_{n\in A} \frac{d^n}{dx^n}\Lambda_{c_n e^{-x}}$. This can be simplified to ...
3
votes
2answers
93 views

No distribution for associated function

Show that there is no distribution $f \in D'(\mathbb{R})$ such that \begin{equation} f(\phi)=\int e^{1/x^2}\phi(x)dx \end{equation} for every $\phi \in C_{0}^{\infty}(\mathbb{R})$ with $supp(\phi) ...
0
votes
1answer
210 views

Limit of distributions of principal value

What is the limit in $D'(\mathbb{R})$ (i.e. in the distribution sense) of \begin{equation} \lim_{t \rightarrow +\infty}\frac{e^{ixt}}{x+i0} \end{equation} where $x+i0=p.v.(\frac{1}{x})-i\pi\delta(x)$ ...
2
votes
1answer
285 views

About a property of the Dirac delta function

How can I show that there is no $u$ satisfying both (i) and (ii):$$(i) \; u \in L^p (\Bbb R^n )$$ and $$(ii) \int_{\Bbb R^n} \delta (x) \phi(x) dx= \int_{\Bbb R^n} u (x) \phi(x)dx\; ( \forall \phi ...
1
vote
1answer
737 views

Inverse function with Dirac Delta

We know that the inverse function of $ y=\log(x) $ is $ y =\exp(x) $. However, what would be the inverse of $ y=\log(x)+ \sum_{n=1}^{\infty}\delta (x-n) $? I have tried with Mathematica, and ...
10
votes
5answers
2k views

Is the Dirac Delta “Function” really a function?

I am given to understand that the Dirac delta function is strictly not a function in the conventional sense and it is a "functional or a distribution". The part which I can not understand why the ...
4
votes
1answer
158 views

Distributional differential equation, somehow related to compact support distributions

I've been mulling over a problem from Friedlander's Introduction to Distribution Theory for a few days now: in Chapter 3 (on distributions with compact support), it asks to solve the differential ...
0
votes
2answers
855 views

example of a function with compact support

Can you give an example of a function which is $C^\infty (\mathbb{R})$ having support on (-1,1) such that $ \int_{-1}^1 f(x)\,dx$=1 and $ \int_{-1}^1 xf(x)\,dx$=0. Thank you.
5
votes
1answer
1k views

Fourier transform of Cauchy principal value

I try to understand the direct computation of the Fourier transform of the distribution `Cauchy principal value' $v.p \frac{1}{x}$. I don't understand the following change of order of integration: $$ ...
3
votes
2answers
441 views

approximating Dirac delta with bounded derivatives

Consider the Dirac delta distribution $\delta$ in $\mathbf{R}^d$. It is quite standard to approximate it by functions $g_n$ with $\|g_n\|_{L^1} = 1$. Is it possible to choose a sequence of test ...
4
votes
1answer
145 views

Generalizing the weak derivative

I am wondering about the weak derivative in time. We say f has a weak derivative f' if $$\int_0^T f\phi' = -\int_0^T f'\phi$$ for all $\phi \in C_0^\infty(0,T)$. This definition uses the $L^2$ inner ...
1
vote
1answer
83 views

How to prove the density result?

How to prove that $C_c^\infty(R^n)$ is dense in $C^0(R^n)$,where the topology of $C^0(R^n)$ defined as follows $u_k\rightarrow 0$ in $C^0(R^n)$ if and only if $\forall K\subset\subset R^n$,$sup_{x\in ...
5
votes
2answers
685 views

Proving the integral of the Dirac delta function is 1

Was wondering if my solution is mathematically accurate enough: The question in the book yields: Derive $$ 1=\int_{-\infty}^{\infty} \delta(x-x_i)\ dx_i $$ From $$ ...
2
votes
3answers
3k views

Heaviside step function fourier transform and principal values

I found the following answer on SE: Fourier transform of unit step? However, it is still not clear to me and maybe somebody could explain it clearer. Problem I have the following in my notes of ...
-1
votes
1answer
239 views

Fundamental solutions of PDEs

I have two questions about solving PDEs. $L$ is an linear differntial operator In the complement of the origin, the equation $LE =\delta$ reduces to $LE = 0$. Why? What can say about solutions of $ ...
-1
votes
1answer
64 views

When Dirac function is in $H^{-m}(R^n)$?

If Dirac function $\delta\in H^{-m}(R^n)$,please give the range of $m$?
9
votes
1answer
360 views

How much can a weak derivative differ from a classical one?

Let $B$ denote the unit ball in $\mathbb{R}^n$ and let $f\in C^1(B\setminus\{0\})\cap L^1(B)$. Denote with $\nabla_c f$ the classical gradient, which is defined in $B\setminus\{0\}$, and denote with ...
6
votes
1answer
291 views

Convolution between two distributions

I want to define the convolution $*$ between two distributions $S$ and $T$. For a test function $\varphi$, can I say: $$\langle S * T, \varphi \rangle \doteqdot \langle S, T*\varphi \rangle $$ where ...
3
votes
1answer
112 views

Equicontinuity and uniform boundedness for “distributions”

Exercise. (Rudin, Functional Analysis, chapter 2, pag. 53). Let us consider the space $$ \mathcal D :=\{f \in C^{\infty}(\mathbb R), \, \text{supp}f\subseteq [-1,1] \} $$ with the topology induced by ...
2
votes
1answer
114 views

Properties of Fourier transform of distributions

For distributions the scaling property, $f(ax) = \frac{1}{|a|} \mathcal{F(\frac{u}{a})}$, of the Fourier transform is no longer true. Is there a source that lists which properties of the Fourier ...
2
votes
1answer
214 views

weak derivative and continuous functions (functionals, distributions)

Suppose $\varphi \in C_c^\infty(0,T; H^1(\Omega))$ is a $H^1(\Omega)$-valued test function (it vanishes at $t=0$ and $t= T$), and $f \in C^1(0,T \times \Omega)$. Let $w \in L^2(0,T;H^1(\Omega)$ with ...
1
vote
0answers
37 views

Function of the incremental ratio tends weakly to a distribution

Let $g:\mathbb{R}^3\to\mathbb{R^2}$ be a continuous function. Suppose that there exists $\Omega$ a neighborhood of $0$ where $Xg, Yg \in L^\infty(\Omega)$, with ...
3
votes
1answer
68 views

Show continuity of a function?

Are there theorems or results to show that if for every $\varphi\in \mathcal{C}_0^\infty(\mathbb{R})$ we have, $$\int_{\mathbb{R}} \varphi^k(x)\mu(dx) \leq C$$ Then $\mu(dx) = f(x)dx$ and $f\in ...
0
votes
1answer
272 views

Questions related to distribution function and its “inverse”

Let $f: \mathbb R^n \to \mathbb R$ be a measurable fucntion. Define $F(t) = \mu \{x:|f(x)| >t\}$ Show that $F$ is nonincreasing and right-continuous (done). Define $F^\star(v)=\inf \{t: F(t)\leq ...
2
votes
1answer
58 views

A particular product of distributions

Suppose you have two continuous functions $f,g: \mathbb{R}\to\mathbb{R}$; is the product $f'g$ as a distribution, at least locally? I am interested in a local result, actually, so you can as well ...
0
votes
1answer
104 views

Composition of a distribution with a map

Suppose $\lambda \in C_c^\infty(\mathbb{R})^*$ is a distribution and $f: \mathbb{R} \to \mathbb{R}$ is a continuous map of the real line. In addition assume $f$ has compact support. How can I make ...
1
vote
1answer
127 views

The Dirac impulse and Fourier transform

Here wikipedia it is said that the Dirac delta could be thought of as $$ \delta(x) = \left\{ \begin{array}{ll} \infty &, x = 0 \\ 0 &, x \ne 0 \end{array}\right. $$ and here that the ...
3
votes
0answers
338 views

Property of derivative of Dirac delta function in $\mathbb{R}^n$

With reference to Property of Dirac delta function in $\mathbb{R}^n$, is there a similar formula for $\langle g^*\delta', f \rangle$ (or even $\langle g^*\delta^{(n)}, f \rangle$)? By similar I mean a ...
1
vote
2answers
197 views

Distributional/weak time derivative basic question

Suppose we have $u \in L^2(0,T;H^1(\Omega))$, and $v \in L^2(0,T;H^{-1}(\Omega))$ is the weak time derivative of $u$, so by definition it satisfies $$\int_0^T u(t)\phi'(t) = -\int_0^T v(t)\phi(t)$$ ...
6
votes
1answer
290 views

Convolution square root of $\delta $

I want to somehow classify the distributional solutions of the equation $$ f \ast f = \delta $$ where $\delta = \delta _0$ is the Dirac delta distribution. Clearly, by Fourier transformation, we ...
2
votes
1answer
240 views

Proof that limit goes to zero without Riemann-Lebesgue lemma

Let $\varphi$ be a test function ($\varphi$ is smooth and has compact support - is zero outside some bounded interval). I know that the following $$ \lim_{\epsilon \to 0_+} ...
1
vote
0answers
71 views

Lebesgue-Stieltjes integral as a generalized function

Given some convex function $f(x)$, $x >0$ we can define a distribution $F \in \mathcal{D}'(0,\infty)$ using Lebesgue-Stieltjes integral $$ \langle F, \varphi \rangle ...
3
votes
1answer
234 views

Second derivative of convex function

Let $f(x)$, $x>0$ be a convex function. Then it's distributional second derivative is defined by the rule $$ \langle f''(x),\varphi(x)\rangle = \langle f(x), \varphi''(x)\rangle $$ for any ...
2
votes
1answer
180 views

Sobolev spaces of infinite order

I do have a question about the Sobolev spaces of infinite order. Let me first define them: Let $H^s(\mathbb{R}^n)$ denote the Sobolev space of order $s \in \mathbb{R}$. We can naturally identify ...
2
votes
2answers
944 views

Principal value of 1/x- equivalence of two definitions

As far as I know, the principal value of a non-summable function like $1/x$, denoted $\mathcal{P}(1/x)$, is a distribution that that acts on some smooth function $f$ in some test-function space and ...
2
votes
2answers
100 views

particular solution by variation of constants

I have this ODE : $y^{''}(x)-Ay(x)=Bx \delta_{0}(x)$ where $A,B$ are constants and $\begin{equation} \delta_{0}= \begin{cases} \infty & \text{if $x=0$}, \\ 0& \text{else}. \end{cases} ...
1
vote
1answer
151 views

How to show that limit is a delta function

Let $\{\phi_{n}(t)\}_{n=1}^{\infty}$ be a complete orthonormal system at $[a,b]$. Then $$ \sum\limits_{n=1}^{\infty} \phi_{n}(t)\phi_{n}(s) = \lim\limits_{N \to \infty} \sum\limits_{n=1}^{N} ...
1
vote
1answer
401 views

Singular support of distributions

I've read the following definition: Let $u \in \mathcal{D}^\prime(\mathbb{R}^n)$. We say that $y_0 \notin \mathrm{sing} \ \mathrm{supp} \ u$ if there exists $\phi \in ...
2
votes
1answer
160 views

Extension of formula for solution of heat equation on distributions

Consider a heat equation $$ u_{t} = u_{xx}+f(t,x),\; (x,t) \in (0,L)\times(0,T] \\ u(0,x) = 0, \; x \in [0,L] \\ u(t,0) = 0, \; t \in [0,T] \\ u(t,L) = 0, \; t \in [0,T]. $$ If $f(t,x)$ is ...
2
votes
0answers
74 views

Does multiplication commute with taking of fundamental solution (heat equation)

Let $\Phi(t,x)$ be a heat function, $$ \Phi(t,x) = \frac{1}{\sqrt{4 \pi t}} \exp\left(-\frac{x^2}{4t}\right). $$ Then $(\partial_{t} - \partial_{xx})\Phi(t,x) = \delta(t)\delta(x)$. Furthermore, ...
2
votes
0answers
81 views

Representation of distribution by nonnegative measure

Let $T \in \mathcal{D}'(\mathbb{R}_{+})$ be a distribution on $\mathbb{R}_{+}$ such that for any $f \in \mathcal{D}(\mathbb{R}_{+})$, $f \geqslant 0$ we have $$ \langle f, T \rangle \geqslant 0 $$ ...
2
votes
0answers
92 views

Weak derivative and homeomorphisms commute

Suppose $\phi:H^1(S) \to H^2(R)$ is a linear homeomorphism, where $S$ and $R$ are compact surfaces in $\mathbb{R}^n$. Let $\varphi \in D(0,T;H^1(S))$ be a $H^1(S)$-valued $C_c^\infty(0,T)$ function. ...
1
vote
1answer
114 views

Convergence of distributions in $L^p$

If I understand correctly, distributions $F_n \in C^\infty_c(\mathbb{R})^*$ are defined based on how they act on test functions $\phi \in C^\infty_c(\mathbb{R})$. What does it mean then to say $F_n ...
1
vote
0answers
30 views

Solve an equality for distribution

I have an equality that holds for any $\lambda > 0$ $$ \int\limits_{0}^{\infty}{e^{-\lambda t^{\alpha}}} T(t)dt = \int\limits_{0}^{\infty}e^{-\lambda t^{\alpha}} dg(t), $$ where $\alpha > 0$ ...
5
votes
4answers
3k views

On the derivative of a Heaviside step function being proportional to the Dirac delta function

I am learning Quantum Mechanics, and came across this fact that the derivative of a Heaviside unit step function is Dirac delta function. I understand this intuitively, since the Heaviside unit step ...
2
votes
0answers
686 views

Distributions supported on a single point

Let $d=1$. (i) Show that if $\lambda$ is a distribution and $n\geq1$ is an integer, then $\lambda x^n=0$ if and only if $\lambda$ is a linear combination of $\delta:=\delta_{\{0\}}$ and its first ...
3
votes
1answer
124 views

What is the use of $H_s$ for non-integer $s$?

So we have the whole set of theory for Sobolev spaces \begin{equation} H_s(\mathbb{R}^d)=\{u\in D'(\mathbb{R}^d):(1+|y|^2)^{s/2}\hat{u}\in\mathcal{L}^2(\mathbb{R}^d)\}, \end{equation} and we know that ...
2
votes
1answer
70 views

Non-regularity of non-elliptic operator

Let $\Omega\subset\mathbb{R}^d$ be open,and $P(D)=\operatorname{\sum_{|\alpha|\le N}}f_{\alpha}D^{\alpha}$ be an elliptic differential operator. Rudin proves in Functional Analysis Part II the ...