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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12
votes
3answers
1k 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
562 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
2answers
756 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 ...
9
votes
3answers
517 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
575 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 ...
9
votes
1answer
173 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
326 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
215 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
144 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
280 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 ...
8
votes
5answers
1k 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 ...
8
votes
2answers
181 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
2answers
783 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
273 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 ...
8
votes
0answers
84 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 ...
7
votes
3answers
5k 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 ?
7
votes
4answers
205 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) \\ ...
7
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 ...
7
votes
2answers
258 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
256 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
318 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 ...
7
votes
1answer
135 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
2answers
797 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
1answer
671 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
0answers
97 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 ...
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
2answers
605 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 ...
6
votes
4answers
266 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
284 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
222 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
94 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
1answer
136 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
275 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; ...
6
votes
1answer
243 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
100 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
185 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 ...
6
votes
1answer
51 views

Which normed space have Fatou's property?

There is tool in mathematics, more specifically in Analysis, which has immense applications in handling delicates estimates, limiting arguments, etc... ; namely Fatou property. Let $E$ be a normed ...
6
votes
2answers
77 views

Is it possible to mathematically explain why solids go under mollification when heated?

Well, I'm sure that many people on MSE might object that this is not a math question, however, I think that there might be a well-posed mathematical answer to this question, or at least I hope so. We ...
6
votes
1answer
231 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 ...
6
votes
0answers
250 views

Idea behind distributional solutions

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...
5
votes
4answers
2k 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 ...
5
votes
4answers
272 views

delta functions $e^{x}\delta (x)=\delta (x)$

How would you prove that; $$e^{x} \delta (x)= \delta (x)$$ Is it anything to do with the following relationship; $$ \int_{-\infty}^{\infty} g'(x)h(x)\,dx = \int_{-\infty}^{\infty} g(x)h'(x)\,dx.$$ ...
5
votes
4answers
479 views

What is the relationship between generalized functions and things like the Riesz representation theorem?

I just watched this video of Prof. Osgood's lecture on Fourier Transforms, and it seems to me that there's some connection between his talk of distributions (generalized functions) and the usual ...
5
votes
1answer
329 views

How should I understand a PDE that contains distribution or measure mathematically?

We know that the Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, $$ \delta(x)=\begin{cases}+\infty, ...
5
votes
1answer
148 views

Computing the integral $ \lim\limits_{\epsilon\to 0} \int_{-2}^{0} \frac{e^{1/x(x+2)}}{x+1+i\epsilon} $

I got stuck when calculating of this expression $$ \lim_{\epsilon\rightarrow 0} \int_{-2}^{0} \frac{e^{\frac{1}{x(x+2)}}}{x+1+i\epsilon} $$ I will be grateful for the advice.
5
votes
2answers
324 views

Square root of compactly supported C-infinity function

Given $u \in \mathcal{C}^\infty_0(\mathbb{R}^n)$, $u \geq 0$ everywhere, is $v(x) = \sqrt{u(x)}$ also in $\mathcal{C}^\infty_0$? It is clear that the only problematic points are the boundary of the ...
5
votes
1answer
168 views

Taylor series and tempered distributions

Suppose we have a function $\psi$ in $\mathbb{R}$ other than a polynomial which is equal to its Taylor Series for every point in $\mathbb{R}$. Why is the following statement valid? When we interpret ...
5
votes
1answer
462 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 ...
5
votes
1answer
240 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 ...