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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2
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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 ...
0
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0answers
29 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 ...
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} - ...
4
votes
0answers
32 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 ...
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 ...
2
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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
34 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?
1
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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 ...
1
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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
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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
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
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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 ...
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. ...
0
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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 $ ...
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 ...
4
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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
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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 ...
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 ...
3
votes
1answer
53 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 $$ ...
2
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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, ...
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
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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 ...
3
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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
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
1
vote
1answer
76 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, ...
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 ...
2
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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
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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
41 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 ...
2
votes
1answer
74 views

Multiplication of a distribution by a continuous function

In my work, I am encountering the issue of having to multiply a continuous function (not necessarily differentiable) by a distribution. It seems to me that if $f(x)$ is a continuous function on ...
1
vote
1answer
28 views

Fourier transform of the principal value distribution

I would like to compute the Fourier transform of the principal value distribution. Let $H$ denote the Heaviside function. Begin with the fact that $$2\widehat{H} =\delta(x) - \frac{i}{\pi} ...
2
votes
1answer
33 views

Weak limits and computing the Fourier transform of the Heaviside function

A common problem on this site is to compute the Fourier transform of the Heaviside function that is $0$ on the negative reals and $1$ on the positive reals. A standard technique is to consider the ...
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0answers
19 views

Distribution which is not a finite sum of derivatives of continuous functions

Recall that fact that a distribution which is compactly supported can be written as a finite sum of derivatives of continuous functions. This fails for general distributions in $D^\prime ...
0
votes
1answer
23 views

Relation between a distribution in $\mathbb{R}^{1}$ and $\mathbb{R}^{2}$

From Chapter 1, Exercise 8 and 9, Strichartz's book: Distribution theory & fourier transforms Suppose $f$ is a distribution on $\mathbb{R}^1$. Show that $<F,\phi> = ...
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0answers
54 views

Solving Linear ODEs over the space of distributions

I am encountering in my work linear differential equations with coefficients that involve things like Heaviside functions and delta Dirac functions. I know how to find things like Green's functions or ...
0
votes
1answer
39 views

1D green function

I want to solve the problem $$ \frac {\partial^2u}{\partial t^2} + \alpha \frac {\partial u} {\partial t} = f(t)$$ $$u(0)=u'(0)=0$$ $$f(t)=exp(-\beta t)$$ Using distributions (Green functions). ...
0
votes
1answer
22 views

If two distributions vanish on the same set of test functions, then one is a constant multiple of the other

From Chapter 1, Exercise 14, Strichartz's book: Distribution theory & fourier transforms Suppose $f$ & $g$ are distributions such that $\langle f,\phi\rangle=0 \Leftrightarrow \langle g,\phi ...
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4answers
37 views

Proof of the construction of Dirac Delta

The Dirac Delta function pops up in a wide variety of applications, especially in applications that require Laplace and Fourier transforms. But my question is: what's the proof that the distribution ...
0
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0answers
31 views

Limit of Distribution, Hilbert Transform

I want to peform a distributional limit of the following distribution: $\frac{2 i}{x^2 \epsilon} e^{-(t + x)^2/(4 \epsilon)} (F(\frac{t - x}{2 \sqrt{\epsilon}}) - F(\frac{t + x}{2 ...
1
vote
0answers
35 views

Show that continuous linear maps on the space of test functions take $C_K^\infty(\Omega)$ into some $C_{K_N}^\infty(\Omega)$

Let $\Omega$ be a nonempty open subset of $\mathbb{R}^n$, and let $\cup_{n=1}^\infty K_n = \Omega$ be an exhaustion of $\Omega$ by compact sets. Let $\mathcal{D}(\Omega) = \mathcal{D}$ be the standard ...