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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4
votes
2answers
129 views

How to prove that $x \rightarrow e^{1/x}$ is not a restriction of any real distribution to $ \mathbb {R}_+$?

This is an excercise 2.2 from Hormander, vol. I: Does there exist a distribution $u$ on $\mathbb{R}$ with the restriction $x \rightarrow e^{1/x}$ to $\mathbb{R}_+$? The answer, provided in the book, ...
2
votes
1answer
39 views

Hankel transform of a Bessel function of different order

Here I found that $$ \int_0^\infty J_\nu(kr) J_\nu(sr) r dr = \frac{\delta(k - s)}{s} = \frac{1}{s^2}\delta\left(1 - \frac{k}{s}\right). $$ I wonder how can that be derived and if a similar method can ...
7
votes
5answers
16k 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 ...
0
votes
0answers
32 views

Can we make $S_n \to \delta_x$ for $S_n$ an exponential polynomial?

Consider $f_\lambda: \Bbb{R}_+ \to \Bbb{R}_+$,$$f_\lambda(t) = e^{-\lambda t}$$ Now consider the finite linear combinations of these functions (exponential polynomials) $$ S(t) = \sum_{i = 1}^N ...
7
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 ...
1
vote
1answer
29 views

Reference for “distributional derivative being zero implies being constant”

I know that if a distribution (generalized function) has zero derivative, then it is a constant. I also know the proof. But I have a hard time finding a reference which contains a statement of this ...
0
votes
1answer
29 views

Integral (Fourier transform) of Heaviside radial function in 3D

I am trying to calculate the following integral: $ \int \frac{d k_x d k_y d k_z}{(2 \pi)^3} \left[ \exp( - \frac{(k_x^2 + k_y^2 + k_z^2) \sigma^2}{2}) + \frac{1}{2} H(\sqrt{k_x^2 + k_y^2 + k_z^2} - ...
1
vote
0answers
51 views

Extending an identity for the Dirac delta function

The identity $$x^p \; \delta^{(n)}(x) = (-1)^p \frac{n!}{(n-p)!} \; \delta^{(n-p)}(x)$$ can easily be derived from the generalized Leibnitz formula for $n$ and $p$ positive integers: $$\int \; x^p ...
4
votes
0answers
33 views

“Contradiction” to Bochner's theorem for distributions

I recently asked a question "For what values of $\lambda$ the distribution $(x-i\epsilon)^{\lambda}$ is positive?". User Marcel was kind enough to point out in his answer that one uses Bochner's ...
2
votes
0answers
32 views

Another equivalent characterization of Schwartz function?

Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that $$ \sup_{x\in\mathbb{R}^n}\left||x|^k\Delta^{p}\psi(x)\right|<\infty $$ for all ...
1
vote
1answer
35 views

The Fourier transform of a tempered distribution is supported at the origin

If the Fourier transform of a tempered distribution $G$ is supported at the origin, does this imply that $G$ is a constant? Can anyone give a reference or a short argument?
2
votes
2answers
42 views

How to take derivatives of a convolution when the kernel's derivative is in the distribution sense?

I came need to take the derivative of the following convolution: $$ \int_{-\infty}^\infty \operatorname{sgn}(x-y)e^{-|x-y|}f(y) \, dy $$ However, the derivative of the kernel only exists in the sense ...
1
vote
0answers
37 views

Distributions and non-smooth functions

Is it possible to define distributions acting on non-smooth functions? The reason I'm asking is because of the rendering equation $$L_o(x,\omega_o) = \int_\Omega f_r(x,\omega_o,\omega_i) ...
2
votes
0answers
20 views

Deriving a certain delta-sequence with respect to its index

At the end of some calculations I've reached $$\lim \limits _{t \to 0_+} \int \limits _{\Bbb R ^n} \frac {h(t,x,y)} t f(y) \Bbb d y$$ where $$h(t,x,y) = \frac {\Bbb e ^{\frac {\Bbb i |x-y|^2} ...
2
votes
1answer
52 views

Why is the support of Dirac distribution $\{0\}$?

Distributions are of two types: those that are obtained from locally integrable functions, and those that aren't. For the first type, the support of distribution is simply the support of the function. ...
1
vote
0answers
48 views

How to show that $\int_{-\infty}^{\infty} \mathrm{d}^3 \textbf{k} \frac {e^{i \textbf{k x}}} {(2 \pi)^3} = \delta^3(x)$ in spherical coordinates?

Recently I had to deal with Fourier transformations and delta functions, and I was wondering how about that. I know, that its trivial to show in cartesian coordinates, but i couldn't do it in ...
3
votes
1answer
40 views

Convolution of tempered distributions where one has compact support.

For $u\in\mathcal E'(\mathbb R^n)$ and $v\in\mathcal S'(\mathbb R^n)$, we defined $u\ast v$ by $\langle u\ast v, \phi\rangle = \langle v, \check u \ast \phi \rangle$ for all $\phi\in\mathcal S(\mathbb ...
0
votes
0answers
26 views

Way to think about weak deriviate

Something hit me when I read the definiton of weak derivite. Would it be right to think about the weak deriviate in terms of distributons, i.e that the distribution $\int f \phi$ induced by f in $ ...
8
votes
0answers
117 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 ...
2
votes
1answer
41 views

Problem in distribution theory and tempered distributions

I just encountered this question in my real analysis class involving distribution theory it is question 25 chapter 9 from Folland's real analysis second edition, which reads as follows: Suppose ...
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 ...
0
votes
1answer
46 views

Understanding Distributional Meanings and Test Functions for PDEs

My question is about distributional meanings in PDEs. My specific question is at the bottom, but I'd be interested in a bit of general theory (even if this is a link to a particularly good set of ...
4
votes
1answer
67 views

Specific problem on distribution theory.

*****Note: Parts A, C and D I managed. Only need help on part B now would really would appreciate the help on B Hi, in my summer real analysis (or measures and real analysis as my instructor refers ...
0
votes
0answers
40 views

Dirac Delta Distribution and non-compactly supported test function

I would like to know if there is any problem with defining the following expression: $$ I = \int_0^\infty g(t) \delta(f(t))\mathrm{d}t $$ where $0<\lim\limits_{t\to\infty} g(t) =L<\infty$ and ...
3
votes
1answer
54 views

Lebesgue integral of Dirac delta

If I recall correctly, for a bounded function $f$ $$ \int_{\mathbb{R}} f \, d\mu = \int_{\mathbb{R} \setminus \{ a \} } f \, d\mu + f(a) \mu (a).$$ For the Lebesgue measure, $\mu(a) = 0$ and $$ ...
1
vote
1answer
37 views

Fourier transform of a $H(x)$ product distribution

So I am given this simple example, where $T \in \mathcal{S}(\mathbb{R})$: \begin{equation} T=(\mu +\lambda x+\beta x^2)H(x) \end{equation} where $H(x)\in \mathcal{S}(\mathbb{R})$ (also notated as the ...
2
votes
0answers
52 views

What are some properties of the sheaf of distributions?

In a course on measure theory, the lecturer proved that distributions (on a locally convex space I think) form a sheaf $\mathcal D$. He isn't interested in sheaves, so he didn't elaborate. Afterwards, ...
3
votes
1answer
511 views

Fourier transforms of the Heavyside function and the absolute value function

I'm trying to obtain these two Fourier transforms. First of all, I'm using the following definition of Fourier transform: $$\cal{F}(f)(y)=\int_{-\infty}^{\infty}f(x)e^{-ixy}\;dx$$ What I have so ...
2
votes
1answer
52 views

Confusion about inclusions of dual spaces

We have the triple inclusions $H^1_0(\Omega)\subset L^2(\Omega)\subset H^{-1}(\Omega)$, where the second inclusion is not literal but in the sense of distributions. Related to this answer, why is the ...
0
votes
1answer
47 views

different generalized functions?

I am trying to solve a PDE that's order 1 in time $t\ge0$ and order 2 in space $x\ge0$. The solution $u(x,t)$ exists, is unique and possesses the following properties: $u(x,t)\ge0$ for all ...
2
votes
1answer
48 views

Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0 $$ Then, to find ''bound states'', you solve on the right and find the ...
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 ?
3
votes
0answers
105 views

No consistent theory can define a product of distributions: why?

I have been told there cannot be a consistent theory defining a distribution product. Googling for information, I found 1 and 2. Number 1 gives interesting hints on what might happen, and defines a ...
2
votes
0answers
36 views

Schwartz impossibility result

I was wondering what made it impossible to define a product of distributions. Googling, I found two questions, one of which stated the following impossibility result: There is no associative ...
2
votes
1answer
44 views

Weak convergence and integrals

Assume $$u_k\rightharpoonup u,\quad v_k\rightharpoonup v\quad\text{in}\quad L^1(0,T;Y)\tag{1}$$ and $$\int_0^T u_k(t)\varphi'(t)\ dt=-\int_0^T v_k(t)\varphi(t)\ dt\tag{2}$$ for some $\varphi\in ...
0
votes
1answer
75 views

Is this operator a Fourier multiplier operator?

I want to study the Fourier transform of $$L_{\alpha}(t) = \frac{e^{i\alpha t}}{t^2} - i\frac{\alpha}{t}$$ Basically i am trying to get a grip on, given a $f$, what is $f(t)\ast L_{\alpha}(t)$ and am ...
5
votes
1answer
65 views

Fourier transform of the Heaviside function

As you can see from the title I want to calculate the Fourier transform of the Heaviside function $u(t)$. Proven the the Heaviside function is a tempered distribution I must evaluate: $$ \langle ...
3
votes
3answers
58 views

Definition in Lax “sequence of continuous functions tending to $\delta$”, are distributions needed for understanding?

I'm trying to read Lax's functional analysis. In chapter 11 he makes a definition which I don't like. A sequence of continuous functions ${k_n}$ on a $[-1,1]$ tends to $\delta$ if $\int_{-1}^{1} ...
0
votes
1answer
56 views

How to make a change of variable inside the Dirac delta?

Let $\delta(\phi) = \phi(0)$ be the dirac delta. I would like to compute $\int_{\mathbb{R}} h(x) \delta(\lambda x) dx$ 1) Since $\delta$ is an unit mass on $0$ $$\int_{\mathbb{R}} h(x) ...
8
votes
1answer
175 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 ...
1
vote
1answer
77 views

Fourier Transform of a Polynomial

Lets say you are given \begin{equation} f(x)=1+x^3 \end{equation} and the definition of Fourier transform: \begin{equation} \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}f(x)dx, ...
8
votes
1answer
119 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 ...
2
votes
0answers
46 views

Why are chains topologically analoguous to distributions?

This question is related to my other question here but is different enough that I thought I might ask separately. At the nLab page on rational homotopy theory it is stated that chains are ...
2
votes
0answers
33 views

What is the connection between $\widehat{\mathbb Q G}$ and distributions near the identity of $G$?

I'm studying Quillen's rational homotopy theory and trying to understand this MathOverflow description of Quillen's functor provided by Hiro Lee Tanaka. When discussing connections between how ...
0
votes
1answer
24 views

Functions and distribution integrals

Suppose $ f, g $ are two smooth functions and that for all $ h: \mathbb{R} \rightarrow \mathbb{R} $: \begin{align*} \int f(h(x)) + g(h(x)) \frac{d^2 h}{dx^2} dx = 0 \end{align*} Can I conclude that $ ...
2
votes
1answer
42 views

ODE's & PDE's: Homogenous piecing vs Eigenexpansion vs Green functions

I don't know if i'm within rules of the forum to ask this question. If i'm not please comment before downvoting. If you know of a source that answers these questions, please suggest. It would be ...
2
votes
0answers
33 views

Check smoothness at point looking at Fourier transform

Let $u \in \mathscr E'(\mathbb R^n)$ we a distribution with compact support. Then $u \in C^\infty(\mathbb R^n)$ if and only if for any $N \in \mathbb N$ there exists $C_N > 0$ such that $$ ...
0
votes
1answer
55 views

Distributions corresponding to $\frac{1}{|x|}$

Stirchartz's book ("A guide to distribution theory and fourier transforms" ) has Chapter 1 exercises Here $\mathcal{D(\mathbb{R}^1)}$ is a set of test functions $\phi:\mathbb{R} \rightarrow ...
9
votes
4answers
280 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) \\ ...
2
votes
2answers
388 views

Delta function with imaginary argument

We have an integral representation for the Dirac delta function as $\delta(x) = \frac{1}{2\pi} \int_{-\infty}^\infty dk e^{ikx}$ . On the other hand, we have for delta function the property: ...