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. $$\...
- 31
5
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3
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Integrating a product of a Dirac delta distribution and its derivative.
I came across an integral in the wild:
$$\int^{+\infty}_{-\infty}dx\int^{+\infty}_{-\infty}dy \, f(x,y) \, \delta(x-y)\,\frac{\partial}{\partial y}\delta(\, \epsilon- |x-y|\,).$$
In the expression, $\...
- 83
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3
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Rigorous treatment of Green's functions?
I'm looking for a reference that treats Green's functions with full mathematical rigor, at a level similar to Rudin's Functional Analysis. In fact, Rudin's book does treat fundamental solutions, which ...
- 8,082
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Solving Duhamel integral using the mean value theorem
In the system theory Duhamel integral is used to determine the response $y\left( t \right)$ of linear time-invariants systems to any input signal $x\left( t \right)$ based-on the system unit impulse ...
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1
answer
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Is the derivative of a monotone function measurable? (Justifying Tonelli in a distribution function identity)
This semester, I am taking a course in Harmonic analysis. My professor recommended that we obtain a copy of "Fourier Analysis" by Javier Duoandikoetxea, as the lectures are based on the book....
- 579
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0
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Structure for distributional solutions of homogeneous ODE
Consider an $n$-order linear ODE, for instant $y^{(n)}=Ly$ with $L$ being a homegeneous linear differential operator of order $(n-1)$.
In the classical theory, for any set of initial conditions $(y(0),...
- 21
5
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1
answer
209
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When is exactly the identity $f(z)\delta(z−a)=f(a)\delta(z−a)$ valid?
If we assume that
$$
\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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0
answers
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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 ...
- 21
2
votes
2
answers
194
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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
5
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3
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206
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Solving the differential equation $y'' + y = \delta$ in the sense of distributions
The differential equation is as in the title. A priori, I know the solutions should be all of the type $H(x)\sin(x) + \alpha \sin(x) + \beta \cos(x)$ with $\alpha, \beta \in \mathbb{R}$, because the ...
- 341
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1
answer
53
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Subharmonic and weak subharmonic
Let $\, \Omega\subset \mathbb{R}^n$.
We say that a function $u\in C(\Omega)$ is subharmonic if
$$ u(x) \le \frac{1}{\omega_n r^n} \int_{B_r(x)}u(y)\,dy \quad \forall \, x\in \Omega, \forall \, 0<r&...
- 165
2
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Does weak* continuity imply a distribution is defined pointwise?
Suppose $T \in D'(\mathbb{R} \times \mathbb{R})$. I am reading a paper that assumes weak* continuity of $T$ in the second variable with the first variable held fixed, so weak* continuity of the map $t ...
- 1,183
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1
answer
72
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How is initial data prescribed for weak solutions of a PDE?
My question is for any time-dependent PDE, but for the sake of the post consider the 1D advection equation:
$$u_t + u_x = 0.$$
A weak solution is a function/distribution $u$ that satisfies
$$\langle u,...
- 1,183
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0
answers
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Discrete even distributions under dilation in S′(ℝ): convergence to delta-jets? [closed]
Let
$$
\Lambda = \sum_j w_j\big(\delta(\cdot - q_j) + \delta(\cdot + q_j)\big) \in \mathcal{S}'(\mathbb{R}),
$$
where $|q_j| \to \infty$ and $w_j$ grow at most polynomially, so that $\Lambda$ is a ...
- 11
6
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1
answer
199
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A Theorem of Mihklin-Hörmander for $n=1$
The following Theorem is proved for all $n\in \mathbb{N}$ in Grafakos (Thm 6.2.7), but I believe that there is an issue in his proof, specifically when he asserts that $$\sum_{j\leq 0} \int_{\delta<...
- 4,047