Questions tagged [distribution-theory]
Use this tag for questions about distributions (or generalized functions). For questions about "probability distributions", use (probability-distributions). For questions about distributions as sub-bundles of a vector bundle, use (differential-geometry).
3,868 questions
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Measure of Lebesgue point
Let's say I have a function $u \in BV(I)$ where $I \subset \mathbb{R}$ is an open interval. Its distributional derivative $Du$ is a Radon measure. Now let's say $x \in I$ is a Lebesgue point i.e. $$\...
- 3,342
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Find the second distributional derivative of $\left|\sin(x)\right|$
I am trying to find the second distributional derivative of $f(x)=\left|\sin(x)\right|$, defined for $x$ in $\mathbb R$.
I have not got far but I started like this:
Let $\varphi$ be defined in $S(\...
- 966
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Integrating a product of a Dirac delta distribution and its derivative.
I came across an integral in the wild:
$$\int^{+\infty}_{-\infty}dx\int^{+\infty}_{-\infty}dy \, f(x,y) \, \delta(x-y)\,\frac{\partial}{\partial y}\delta(\, \epsilon- |x-y|\,).$$
In the expression, $\...
- 187k
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Fourier series convergence in $C^\infty(\mathbb{R}^N)$
I have problems with this lemma, page 45 of the book "Introduction to the theory of distributions" by Friendlander and Joshi.
$Lemma$. Let $I=(0,1)^N$ be the unit cube in $\mathbb{R}^N$ with $N>1$....
- 15.3k
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Rigorous treatment of Green's functions?
I'm looking for a reference that treats Green's functions with full mathematical rigor, at a level similar to Rudin's Functional Analysis. In fact, Rudin's book does treat fundamental solutions, which ...
- 6,267
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Solving Duhamel integral using the mean value theorem
In the system theory Duhamel integral is used to determine the response $y\left( t \right)$ of linear time-invariants systems to any input signal $x\left( t \right)$ based-on the system unit impulse ...
- 10.6k
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Is the derivative of a monotone function measurable? (Justifying Tonelli in a distribution function identity)
This semester, I am taking a course in Harmonic analysis. My professor recommended that we obtain a copy of "Fourier Analysis" by Javier Duoandikoetxea, as the lectures are based on the book....
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When is exactly the identity $f(z)\delta(z−a)=f(a)\delta(z−a)$ valid?
If we assume that
$$
\delta(z-1/2)e^{2\pi inz} = \delta(z-1/2)e^{\pi in} = \delta(z-1/2) (-1)^n,
$$
where we used the Dirac delta. This implies
$$
\delta(z-1/2) \sum_{n\in\mathbb Z} e^{2\pi inz} = \...
- 2,341
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Structure for distributional solutions of homogeneous ODE
Consider an $n$-order linear ODE, for instant $y^{(n)}=Ly$ with $L$ being a homegeneous linear differential operator of order $(n-1)$.
In the classical theory, for any set of initial conditions $(y(0),...
- 21
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Local base for inductive limit topology
The following is based on Schaefer & Wolff's Topological Vector Spaces, 2nd edition, §II.6. Let $E$ be a vector space, $\{E_\alpha\}_{\alpha\in A}$ be locally convex spaces (not necessarily ...
- 8,082
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Prove that this double sum gives delta funtion in the limit
I stumbled upon this while solving a quantum mechanics problem.
Consider the following expression ($-L<x,x^\prime < L$):
$$f(x-x^\prime)=\frac{1}{2L}\lim_{N\to\infty}\frac{1}{N} \sum_{n,m=-N}^{N}...
- 24.3k
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How to show that $\delta^3(\vec{x}-\vec{a})=\lim_{\alpha\to0} \frac{\alpha/\pi^2}{(\alpha^2+|\vec{x}-\vec{a}|^2)^2}$
I hope to show that:
$$\delta^3(\vec{x}-\vec{a})=\lim_{\alpha\to0} \frac{\alpha/\pi^2}{(\alpha^2+|\vec{x}-\vec{a}|^2)^2}$$
I want to show by:
$$\int^{+\infty}_{-\infty} f(\vec{x}) \lim_{\alpha\to0} \...
- 30.7k
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Prove that the distributional derivative of the Cantor function is the the $\log_3 2$-Hausdorff measure restricted to the Cantor set
The Cantor function has weak derivative equal to $0$ a.e.
Its distributional derivative should be the $\log_3 2$-Hausdorff measure restricted to the Cantor set, but I'm having troubles doing the ...
CommunityBot
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How to show for a distribution $T$ and a test function $\varphi,~~T'[\varphi]\equiv -T[\varphi']\;?$ [closed]
For a generalized
function $T,$ we define
$$T'[\varphi] ~≡~ −T[φ']~~~~~~\forall φ ∈ \mathcal D(Ω).$$
where $\mathcal D(\Omega)$ denotes the test function space.
I'm not getting how they ...
- 11k
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Trying to evaluate the limit $\lim_{\epsilon\to 0}\int d\Omega \sin^2\theta \phi(\epsilon,\Omega)$ over the sphere of radius $\epsilon$.
I'm trying to evaluate the action of a certain distribution, and have encountered the following limit:
$$\lim_{\epsilon\to 0}\int d\Omega \sin^2\theta \phi(\epsilon,\Omega),$$ where $\epsilon$ is the ...
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