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).

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17
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2answers
2k 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 ...
13
votes
1answer
106 views

Normal form of currents

Let $M$ be an $n$-dimensional manifold. Then the space of currents $\mathcal D^k(M)$ of degree $k$ on $M$ is the space of continuous linear functionals on the space of test $n-k$-forms. Two typical ...
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
5answers
3k 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 ...
11
votes
5answers
26k 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 ...
11
votes
3answers
1k 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.
11
votes
2answers
285 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: ...
11
votes
2answers
292 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
3answers
1k 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 ...
10
votes
4answers
332 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) \\ ...
10
votes
1answer
1k 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
527 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 ...
10
votes
2answers
756 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 ...
10
votes
0answers
266 views

Two possible definitions of “vector-valued distribution”

Let $X$ be a reflexive Banach space. Define $$\tag{1} \mathcal{D}^\star(0, T; X)=\left\{ u\colon \mathcal{D}(0, T)\to X\ \text{linear and continuous}\right\} $$ where the topology on the space of ...
10
votes
1answer
210 views

Proving that a family of functions converges 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 ...
9
votes
4answers
10k 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 ?
9
votes
2answers
2k 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 ...
9
votes
1answer
225 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 ...
9
votes
2answers
266 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
227 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
438 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
2answers
197 views

Is $\int |x\rangle \langle x|dx$ Mathematical?

I am enrolling in a Quantum Mechanics class. As we all know, the formulation of the basic ideas from QM relies heavily on the notion of Hilbert Space. I decide to take the course since it might help ...
9
votes
1answer
227 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 ...
9
votes
0answers
178 views

Half Solved: A problem on the heat operator not being elliptic with a weakened version of elliptic regularity

I should first mention this: in my studies of Sobolev spaces I have completed all the questions of chapter 9 from Folland's real analysis with the help of this site and this is my last one, which is ...
9
votes
0answers
144 views

What plays the role of the identity for the generalized convolution associated to the Fourier-Bessel transform?

In traditional Fourier theory, the Dirac delta plays the role of an "identity" for the $L^1$ algebra with respect to the usual convolution. The convolution is traditionally built out of group ...
9
votes
0answers
417 views

If a Radon measure is a tempered distribution, does it integrate all Schwartz functions?

The question might at first sight sound like the answer is trivially "yes", so let me clarify the question a bit. Consider given a Radon measure $\mu$ on $\mathbb{R}^n$. Let ...
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
2k 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 ...
8
votes
4answers
1k 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) ...
8
votes
2answers
234 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
92 views

Can a “continuous” convex combination not be element of the convex hull?

Short version of question: can a "continuous" convex combination not be element of the convex hull? I am not a mathematician, so please excuse me if I am not precise. I consider first, e.g., 4 ...
8
votes
1answer
659 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
949 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 ...
8
votes
6answers
2k views

Distribution theory book

I'm looking for a good book on distribution theory (in the Schwartz sense), I have the basic knowledge as given in Grafakos' Classical Fourier Analysis, but I want to know more about it. Is the ...
8
votes
1answer
497 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; ...
8
votes
2answers
1k 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
216 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
134 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
286 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
4answers
3k views

How to prove that the derivative of Heaviside's unit step function is the Dirac delta?

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 ...
7
votes
2answers
673 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
2answers
166 views

Dirac's delta in 3 dimensions: proof of $\nabla^2(\|\boldsymbol{x}-\boldsymbol{x}_0\|^{-1})=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$

If $T_f$ is a distribution, i.e. a linear functional, continuous according to the convergence defined here, defined on the space $K$ of the functions of class $C^\infty$ that are null outside a ...
7
votes
1answer
309 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 ...
7
votes
1answer
308 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
870 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
211 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
145 views

Only two parts left : Problem on Fourier Transform and convergence of Tempered Distributions

I recently met this problem from Folland's real analysis second edition involving a specific question on distributions (exercise 19 page 299) which reads as follows: On $ R $ let $ F_0 = ...
7
votes
1answer
916 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
8k views

Proof of Dirac Delta's sifting property

A common way to characterize the dirac delta function $\delta$ is by the following two properties: $$1)\ \delta(x) = 0\ \ \text{for}\ \ x \neq 0$$ $$2)\ \int_{-\infty}^{\infty}\delta(x)\ dx = 1$$ I ...
7
votes
3answers
393 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 ...