Questions tagged [distribution-theory]

Question 1

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 ...

Question 2

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 ...

Question 3

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 ...

Question 4

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 ...

Question 5

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 ...

Question 6

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 ...

Question 7

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-...

Question 8

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)\...

Question 9

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

Question 10

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(\...

Question 11

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:...

Question 12

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 ...

Question 13

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

Question 14

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, \, \...

Question 15

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 ...

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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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