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Question 1

0votes

0answers

95views

Is there still active research in the mathematical theory of distributions?

I apologize for the vague question, but I'm genuinely curious about whether research is still being done on distribution theory---is it a well-established tool or are there still relevant open ...

Jotazuma

162

askedNov 23 at 15:49

Question 2

2votes

0answers

124views

Taylor series using distributional derivatives

I recently learned about the theory of distributions. I'm wondering if it's possible to approximate some compactly supported continuous function $f$ on $\mathbb{R^n}$ using the multivariate Taylor ...

Ray

45

askedNov 6 at 16:40

Question 3

2votes

0answers

67views

What are the higher order (distributional) gradients of an indicator function?

Let $\Omega \subset \mathbb R^d$ be a bounded set with sufficiently nice boundary and let $$\mathbb I _\Omega(x):= \begin{cases}1 &x\in \Omega,\\ 0& \text{else}\end{cases}$$ be its indicator ...

Espen Xylander

94

askedNov 6 at 15:28

Question 4

2votes

0answers

65views

Question on Theorem on necessary and sufficient conditions for multiplier operator

In Grafakos' Fundamentals of Fourier Analysis, Theorem 2.8.5 states the following:Let $u \in \mathcal{S}^\prime$ (tempered distributions) and $T(\phi) = \phi \ast u \in \mathcal{S}$. Then $T$ admits ...

trapezoid

21

askedOct 27 at 8:46

Question 5

0votes

1answer

51views

Test functions are zero on boundary of bounded set

Test functions in $C_0^\infty(\Omega) = \{ \varphi:\Omega \to \mathbb{R}\ |\ \varphi \in C^\infty(\Omega) \text{ and } supp(\varphi) \text{ is a compact subset of } \Omega\}$, where $\Omega \subseteq ...

Roberto Cavicchioli

45

askedOct 24 at 16:13

Question 6

1vote

1answer

63views

Extending Test Functions by $0$ is a Topological Embedding

Let $\omega \subseteq \Omega \subseteq \mathbb{R}^n$ be open subsets, and write $\mathscr{D}(\Omega)$ for the space of test functions (compactly supported smooth functions) on $\Omega$ with the ...

AJ LaMotta

99

askedOct 7 at 13:44

Question 7

7votes

2answers

658views

Why must the highest derivative term in an ODE absorb a Dirac delta impulse?

In modeling systems with impulsive inputs, the Dirac delta function often appears on the right-hand side of an ordinary differential equation (ODE), such as:$$a_n\frac{d^ny}{d x^n}+a_{n-1}\frac{d^{n-...

MathArt

1,750

askedOct 5 at 18:06

Question 8

1vote

1answer

90views

Schwartz kernel mapping

Let's consider a Schwartz kernel as a kernel as defined by the Schwartz kernel theorem. I'm assuming one can write it in terms of an integral$$ \langle K, \phi\otimes\psi\rangle = \int K(u,v)\phi(u)\...

flippiefanus

216

askedOct 4 at 5:09

Question 9

2votes

1answer

87views

How can the delta function be expressed as in the Schwartz structure theorem? [duplicate]

On pages 34–35 of Streater & Wightman's PCT, Spin and Statistics, and All That, they sayIt can be shown that every tempered distribution can be written in the form$$T(f) = \sum_{0 \leqslant |k| ...

WillG

7,769

askedSep 25 at 23:55

Question 10

0votes

1answer

101views

Closed form expression for the Inverse Fourier transform of products of (Spherical) Bessel functions

I am interested in the following (inverse) Fourier transform of a function involving a product of spherical Bessel functions:$$\mathcal{I} \equiv \frac{1}{2\pi}\int d\omega e^{-i\omega(t-t_0)}~I(\...

newtothis

49

askedSep 22 at 15:35

Question 11

4votes

0answers

87views

Delta function in Minkowski space

In $3$-dimensional Euclidean space, one can show that the delta function centered at the origin is spherically symmetric in its argument, resulting in the following expression in spherical coordinates:...

Tob Ernack

5,409

askedSep 15 at 12:21

Question 12

1vote

0answers

100views

Derivative of Dirac Delta distribution over variable interval

Consider the following expression$$I(x) =\int_{x}^{0}f(y)\delta'(y-x)dy, \tag{1}$$where $f(x)$ is some (say smooth) function. From partial integration the term $f(x)\delta(0)$ appears, which is ...

ICOR

33

askedSep 10 at 13:13

Question 13

5votes

1answer

72views

Convolution of distributions on the torus is separately continuous

$\newcommand{\mcD}{\mathcal{D}}\newcommand{\R}{\mathbb{R}}\newcommand{\T}{\mathbb{T}}$Let $\mcD'(\T)$ denote the space of distributions on the torus $\T$, i.e., the topological dual of $C^\infty(\T)$ ...

Shtab Shtab

295

askedSep 8 at 1:57

Question 14

0votes

0answers

38views

"Partial evaluation" analogue for distributions

Consider an $n$-ary function $f \colon \mathbb R^n \to \mathbb R$. It is obvious this defines a canonical $(n-1)$-ary function $g$ via partial evaluation$$g(x_1, \, \cdots, x_{n-1}) := f(x_1, \, \...

Markus Klyver

2,582

askedSep 6 at 18:29

Question 15

2votes

1answer

107views

Using n-dimensional fourier transform to solve n-dimensional PDE

I'm not very experienced in using Fourier transforms to solve PDEs, in fact, I'm trying to learn right now. Here is my issue. I'm trying to understand an example found in some lecture notes. Suppose ...

Luke__

492

askedSep 2 at 17:17

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Stack Exchange Network

Questions tagged [distribution-theory]

Is there still active research in the mathematical theory of distributions?

Taylor series using distributional derivatives

What are the higher order (distributional) gradients of an indicator function?

Question on Theorem on necessary and sufficient conditions for multiplier operator

Test functions are zero on boundary of bounded set

Extending Test Functions by $0$ is a Topological Embedding

Why must the highest derivative term in an ODE absorb a Dirac delta impulse?

Schwartz kernel mapping

How can the delta function be expressed as in the Schwartz structure theorem? [duplicate]

Closed form expression for the Inverse Fourier transform of products of (Spherical) Bessel functions

Delta function in Minkowski space

Derivative of Dirac Delta distribution over variable interval

Convolution of distributions on the torus is separately continuous

"Partial evaluation" analogue for distributions

Using n-dimensional fourier transform to solve n-dimensional PDE

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