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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15
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2answers
1k views

Sobolev space is an algebra

How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have ...
12
votes
3answers
2k views

Delta function integrated from zero

I am trying to understand the motivation behind the following identity stated in Bracewell's book on Fourier transforms: $$\delta^{(2)}(x,y)=\frac{\delta(r)}{\pi r},$$ where $\delta^{(2)}$ is a ...
11
votes
3answers
767 views

Dirac Delta or Dirac delta function?

Is Dirac delta a function? What is its contribution to analysis? What I know about it: It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.
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 ...
10
votes
1answer
845 views

Tensor products of functions generate dense subspace?

Let $X$ and $Y$ be two spaces in certain category, $F(\cdot)$ a functor associating each space with a function space (with certain topology). Assume that for any $f\in F(X)$ and $g\in F(Y)$, $f\otimes ...
10
votes
2answers
282 views

Is this sequence bounded ? (An open problem between my schoolmates !)

Let $f$ be a smooth function (say $\mathcal{C}^{\infty}$) in its two real variables ($t$ and $T$). I consider the following sequence defined by $$A_n:=\lim_{T \to \infty} \int_{0}^{1} e^{-n t} ...
10
votes
2answers
566 views

Distributions on manifolds

Wikipedia entry on distributions contains a seemingly innocent sentence With minor modifications, one can also define complex-valued distributions, and one can replace $\mathbb{R}^n$ by any ...
9
votes
4answers
275 views

Iterated Limits Schizophrenia

Consider the functions $g_n(x)$, with $n\in\mathbb{N}$, $n \ge 1$ and $x\in\mathbb{R}$, defined as follows: $$ g_n(x) = \begin{cases} 2n^2x & \text{if }0 \le x < 1/(2n) \\ ...
9
votes
3answers
816 views

Rigorous derivation/explanation of delta function representation?

I am interested in a derivation of the following representation for the Dirac delta function: $$\delta(x-a)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i p (x-a)}dp$$ It is clear to me how the property ...
9
votes
1answer
182 views

Oscillatory integral giving me the willies

So now that my term's over, I've been brushing up on my quantum field theory, and I came across the following line in my textbook without any justification: ...
9
votes
1answer
415 views

Origin of the name 'test functions'

This is a very simple question really: where did the name 'test functions', used nowadays when speaking of infinitely differentiable and compactly supported functions, come from? More to the point: is ...
9
votes
2answers
237 views

How do different notions of “distribution” relate to one another?

In reading "Real Analysis: Modern Techniques and Their Applications" (Folland), I've come across a few different notions of "distribution" or "distribution functions." The distribution function of a ...
9
votes
2answers
178 views

Various kinds of derivatives

Let $f\colon \mathbb{R}\to \mathbb{R}$ be a measurable function. Let us introduce the following notions of "derivative" of $f$. Classical derivative. The unique function $f'_c$ defined pointwise by ...
9
votes
1answer
362 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 ...
9
votes
1answer
185 views

Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$ It is well-known that $\varphi$ is ...
8
votes
4answers
7k views

what is product of delta function with itself?

what is the product of delta function with itself ? what is the dot product with itself ?
8
votes
3answers
2k views

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

How does one prove the following identity? $$\int _Vf(\pmb{r})\delta (g(\pmb{r}))d\pmb{r}=\int _S\frac{f(\pmb{r})}{|\text{grad} g(\pmb{r})|}d\sigma$$ where $S$ is the surface inside $V$ where ...
8
votes
2answers
212 views

A Differential operator.

What are the fundamental solutions for the operator $$\mathcal D=a{\partial^2\over\partial x_1^2}+b{\partial^2\over\partial x_2^2}$$ on $\Bbb R^2 $ with standard cordinates $(x_1,x_2)$. Here ...
8
votes
2answers
1k views

When can one expect a classical solution of a PDE?

When solving a PDE, there may be a classical solution or a weak solution (or distribution solution). But I am wondering that when people talk about "finding a solution" to some PDE, what do they refer ...
8
votes
1answer
569 views

Sequential and topological duals of test function spaces

Given a test function space, in particular $\mathcal{S}=\mathcal{S}(\mathbb{R}^n)$ (the Schwartz space) or $\mathcal{D}=\mathcal{D}(\mathbb{R}^n)$ (the space of compactly supported smooth test ...
8
votes
2answers
920 views

How to prove that the Cantor ternary function is not weakly differentiable?

I am using the standard cantor ternary function $f$ here, as cited in this Wikipedia page. It is an example of continuous, monotone increasing, but not strictly monotone increasing function with zero ...
8
votes
1answer
115 views

Why do we give $C_c^\infty(\mathbb{R}^d)$ the topology induced by all good seminorms?

Briefly, my question boils down to the following: What benefits do we gain from considering the space of test functions in the topology induced by all "good" seminorms, as opposed to other topologies ...
8
votes
1answer
164 views

Do Electrical Engineering Researchers Usually Know Higher Math? e.g. Measure and Distribution Theory, Functional Analysis

I am an EE undergraduate student, and I recently took two courses in signals and systems. We were exposed to things like the Dirac delta (without mention of distributions), Fourier series (without ...
8
votes
1answer
117 views

Delta distributions with nonlinear arguments

I am confused by the use of nonlinear arguments with the Dirac $\delta$ distribution that I am encountering in the literature. This looks like a widespread use, but for concreteness let us focus on a ...
8
votes
1answer
277 views

What is the name for the archetypical example of a test function, $\varphi(x)=e^{1/(x^2-1)}$?

$$ \varphi(x)=e^{1/(x^2-1)} $$ This function (on the interval $\quad]\!-1,1[ \,\,\, $, outside of it simply $\equiv0$) is used as the typical example of a test function / bump function, I have so ...
7
votes
5answers
15k views

Derivative of a Delta function

I know questions similar to this one have been asked, but there is a particular aspect that I'm confused about that wasn't addressed in the answers to the other ones. I'm dealing with an expression ...
7
votes
2answers
454 views

How do we define the $L^p$ norm of a tempered distribution?

I am finishing up a class on function theory and I am trying to reconcile a few statements that I have seen. Let us define $L^p(\mathbb R^n)$ to be the set of measurable functions $f$ so that ...
7
votes
1answer
286 views

Questions about Fubini's theorem

I learned the following from Hunter's Applied Analysis. Denote the Schwartz space $${\mathcal S}({\mathbb R}^n):=\{\varphi\in C^{\infty}({\mathbb R}^n):\sup_{x\in{\mathbb ...
7
votes
2answers
748 views

Topologies on the space $\mathcal D'(U)$ of distributions

In my analysis lecture I am given a topology on the space of distributions as follows: Let $u_k$ be a sequence in $\mathcal D'(u)$, $u \in \mathcal D'(u)$. We say $u_k \rightarrow u$, if ...
7
votes
1answer
179 views

Fourier transform using principal value

Can anyone help me compute the Fourier transform of $ 1/|x|^{n-\alpha} $ in $\mathbb{R}^n $ where $ 0 < \alpha < n $ ? Somehow it becomes the principal value of $ 1/|x|^\alpha $ which I can't ...
7
votes
1answer
781 views

Meaning of polynomially bounded

I am currently reading through some lecture notes on Fourier Transform and Distributions on my own, and came upon the notion of a polynomially bounded function. I am not sure I understand this ...
7
votes
2answers
867 views

Dual of Sobolev space $W^{1,p}(U)$ for $U$ an arbitrary subset of $\mathbb R^n$

this question may be shameful, but nevertheless I can't help myself. Let $U \subset \mathbb R^n$ be arbitrary, in particular not the whole of the space itself. I wonder about the dual of the space ...
7
votes
3answers
252 views

Distribution theory and differential equations.

How does distribution theory plays role in solving differential equations? This question might seem to be very general. I will try to explain, please bear with me. I understand, distributions make it ...
7
votes
1answer
381 views

What's the Fourier transform of these functions?

The Fourier transform of $|x|^{\alpha}$. This is the Fourier transform of a homogeneous function, and there are several cases of various $\alpha$: when $a\leq -n$, it's not a temperate distribution; ...
7
votes
0answers
113 views

In the space of distributions, how big is the subspace of functions?

I'm teaching Distribution theory and many of my students still believes that there is only one kind of distribution : the distribution that can be identified to a $L^1_{\text{loc}}$ function. And I ...
7
votes
0answers
293 views

Connection between distributional and renormalized solutions for Boltzmann equation

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read through some papers by DiPerna and Lions concerning the Cauchy Problem ...
7
votes
1answer
217 views

Delta function and integrating over level sets?

Consider the three-dimensional integral $$ \int_{\mathbb R^3} d^3x\,f(x)\delta(g(x)) $$ where $\delta$ is the dirac delta, $f,b:\mathbb R^3\to\mathbb R$ and $g(x) = 0$ on some surface $S$. Is there ...
7
votes
1answer
98 views

Proving that a family of functions limits to the Dirac delta.

For each $\epsilon > 0$, define $f_\epsilon:\mathbb R\to \mathbb R$ as follows: \begin{align} f_\epsilon(k) = \frac{1}{\pi}\frac{\epsilon}{\epsilon^2+k^2}. \end{align} How does one rigorously ...
6
votes
3answers
2k views

How to prove $\frac{d\theta}{dx} = \delta(x)$?

Here is a problem from Griffith's book Introduction to E&M. Let $\theta(x)$ be the step function $$\theta = \begin{cases} 0, & x \le 0, \\ 1, & x \gt 0. \end{cases} $$ The ...
6
votes
3answers
644 views

Derivatives distribution

Let $f$ be a distribution on $\mathbf{R}^n$ (in the Schwartz sense) such that $$\frac{\partial f}{\partial x_i} = 0 \text{ for $i = 1, \ldots, n$.}$$ Then how to prove that $f$ is a constant? I had ...
6
votes
2answers
1k views

Laplacians and Dirac delta functions

It is often quoted in physics textbooks for finding the electric potential using Green's function that $$\nabla ^2 \left(\frac{1}{r}\right)=-4\pi\delta^3({\bf r}),$$ or more generally $$\nabla ...
6
votes
1answer
182 views

How to make sense of Fourier series for a distribution?

In particular if I have an array of numbers say, $\{c_m\}_{m\in\mathbb{Z}^n}$. Under what conditions can we say that these are the Fourier coefficients of a distribution? [For examples Bessel's ...
6
votes
4answers
577 views

Dirac delta of nonlinear multivariable arguments

How does one compute a dirac delta function with a multivariable argument? For example, compute: $$ \int^{\infty}_{-\infty}{\rm d}x\,{\rm d}y\, \delta\left(x^{2} + y^{2} - 4\right) ...
6
votes
2answers
359 views

Paley-Wiener type theorems for distributions?

In general a theorem of Paley-Wiener type gives a relation between the decay of a function and the smoothness of its Fourier transformation, and there are plenty of them since there are many kinds of ...
6
votes
1answer
231 views

Difficulties in solving a PDE problem

This is an exercise in "Variation et optimisation des formes", chapter 3, Ex. 3.8. The preliminaries are: $$D=(0,1)^2,\ f \in L^2(D),\ x_{ij}=(i/n, j/n),\ 0<i,j<n,$$ $$\Omega_n = D\setminus ...
6
votes
1answer
295 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 ...
6
votes
1answer
143 views

What is $\Delta\frac{1}{|\mathbf{x}|^2}$, as a distribution?

$\newcommand{\x}{\mathbf{x}}$Let $\x$ denote a vector in $\mathbb{R}^3$, $|\x|$ its magintude and $\Delta=\frac{\partial^2}{\partial x 2}+\frac{\partial^2}{\partial y 2}+\frac{\partial^2}{\partial z ...
6
votes
1answer
252 views

What is wrong with my `proof'?(solved)

The question is: Let $k\in C^{0}(\mathbb{R}^{n}-\{0\})$ be a function such that $$k(xt)=t^{-n}k(x)$$ for $0\not=x\in\mathbb{R}^{n},t>0$. Show that the principal value $$\int ...
6
votes
1answer
114 views

Uniqueness of Ordinary Differential Equations in $D^{'}$, the space of Schwartz distribuitions

Let $m \in \mathbb{N}$. For $k=1,...,m$ let $a_k : \mathbb{R} \rightarrow \mathbb{C}$ be a $C^{\infty}$ function. And suppose that: $a_m(x) \neq 0 \; \forall x \in [x_0, \infty[$ And let P be the ...
6
votes
1answer
191 views

Showing that $\int_0^1 x^{\lambda} [ \: \phi(x) - \phi(0)\: ] dx$ is convergent for $\lambda > -2$

Id' appreciate help understanding why the integral $$ \int_0^1 x^{\lambda} [ \: \phi(x) - \phi(0)\: ] dx $$ is convergent provided $\lambda > -2$, where $\phi \in \mathcal{D}(\mathbb{R})$. To ...