# 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).

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### A brick sliding in an horizontal plane after an initial push (under Coulomb's dry friction) - closed form solutions validation?

A brick sliding in an horizontal plane after an initial push (under Coulomb's dry friction) - closed form solutions validation? Posted later after comments: In summary, I am trying to understand what ...
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### $L^p(\mathbb{T})$ is identified with tempered distibution (periodic)?

i know that, the space $L^p(\mathbb{R}^n)$ canbe idenfitied with a tempered distributuion. Can be the space $L^p(\mathbb{T})$ identified with a tempered distribution (periodic?)?
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### CLT for exchangeable and symmetric dependent Bernoulli variables

Consider a finite set of $n$ dependent Bernoulli random variables $X_1$, $X_2$, ..., $X_n$ that are symmetric and therefore exchangeable. This means: Each variable $X_i$ has the same probability of ...
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Here is the proof of Lemma 4.10. On $\mathbb{R}^{1+3} \backslash\{0\}$, we have the identity $$\delta_0\left(t^2-|x|^2\right)=\frac{1}{2 \sqrt{2} t} \mathrm{~d} \sigma_{C_0^{+}}(t, x)$$ where $C_0^{... • 623 3 votes 2 answers 74 views ### Differentiating Dirac delta with product rule I have here an equation. $$h'(t_2) \delta(t_1 - t_2) = [h(t_2) - h(t_1)] \delta'(t_1 - t_2)$$ I checked the equality by integrating both sides with a test function. $$\int d t_1 \phi(t_1) \ldots \... • 1,108 0 votes 0 answers 30 views ### The L^\infty boundedness of the resolvent of Harmonic Oscillator in terms of seminorms I am reading Watanabe's book "Lectures on Stochastic Differential Equations and Malliavin Calculus". In Page 48, it is said that by means of (damped) harmonic oscillator 1+|x|^2-\Delta, we ... • 313 1 vote 0 answers 40 views ### Distribution induced by a Radon measure Let \Omega \subset \mathbb{R}^N be open, and consider a distribution T\in \mathcal{D}’(\Omega). I should prove the following statement. The distribution T is a linear combination of Radon ... • 133 1 vote 0 answers 45 views ### DiracDelta/x = -DiracDelta'? - Use and Correctness of Statement [duplicate] One property of the Dirac Delta Distribution is x \delta'(x) = -\delta(x) because of \int x \delta'(x) f(x) dx = -\int \delta(x) (xf(x))' dx = -\int \delta(x) (f(x)+xf'(x)) dx = -\int \delta(x) f(x)... • 147 0 votes 0 answers 36 views ### Derivation of sifting property of Dirac Delta Distribution from Dirac sequence, one property isn't used I want to start from the properties of the dirac sequence to derive its defining characteristic. To show: \int_{-\infty}^\infty \delta(x) f(x) dx =f(0) Dirac sequence \delta(x)_{n \in \mathbb{N}}: ... • 147 1 vote 1 answer 31 views ### Laplace Transform of Derivative by Limit and Dirac Delta Distribution The Laplace Transformation of the Derivative is given by \mathcal{L}[f'(x)](s) = \int_0^\infty f'(x) e^{-s x} dx = s \mathcal{L}[f(x)](s) - f(0), which can be easily shown by partial integration. I ... • 147 0 votes 1 answer 35 views ### Why the test function space is countable union of \mathcal{D}_{K_i} In the Rudin's functional analysis, in remark 6.9, It says there is countable collection of sets K_i \subset \Omega such that \mathcal{D}(\Omega)=\cup \mathcal{D}_{K_i} \mathcal{D}(\Omega)is the ... • 709 1 vote 0 answers 23 views ### Tempered distribution from locally integrable function My professor said that if a function is f \in L^1_{\text{loc}}\left(\mathbb{R}\right) . Then its associated distribution : T_f is not a tempered distribution , we defined a tempered distribution ... 0 votes 0 answers 50 views ### What is the meaning of this definition? Let \mathscr{S}(\mathbb{R}^{d}) the Schwartz space of rapidly decreasing functions and \mathscr{S}'(\mathbb{R}^{d}) the space of tempered distributions. Let f be such that f^{\gamma} \in C^{\... 0 votes 0 answers 43 views ### Does the Heaviside function belong to W^{s,p} for some s>0? Let us consider n-fold tensor product of Heaviside functions: $$H(x_1, \cdots, x_n) := \prod_{i=1}^n \chi_{[0,\infty)}(x_i)$$ Then, for any bounded open set \Omega \... • 7,829 0 votes 0 answers 31 views ### Understanding different norms in the p-Wasserstein distance The generalized p-Wasserstein distance, for p\geq 1, is given by$$d_W(Q_1,Q_2):=inf \left\{\int_{\Xi_2}||\xi_1-\xi_2||^p \Pi(d\xi_1,d\xi_2)\right\}$$where \Pi is the joint distribution of \xi_1... • 99 1 vote 0 answers 33 views ### How to prove that any Schwarz distribution is can be represented by hyper-function? As far as I understand hyper-functions are generalizations for distributions (at least in 1-dimensional case and for some "good enough" definition of distribution...). What is the simplest ... • 360 2 votes 1 answer 75 views ### Bound of a compactly supported distribution u: \mathrm{C}^{\infty}(\Bbb R) \rightarrow \mathbb{R} is a linear map$$u(\varphi):=\sum_{j=1}^{\infty} \frac{\varphi\left(\frac{1}{j}\right)-\varphi\left(-\frac{1}{j}\right)}{j}$$The question is ... • 3,047 1 vote 1 answer 66 views ### Positive integral appears negative after applying the convolution theorem I am Fourier transforming momentum representation of quantum mechanical wave functions to the position representation. In the weak sense (tempered distribution) we have$$\int_{\mathbb{R}^3} d^3k \ e^{... • 514 4 votes 1 answer 121 views ### How are polynomials of tempered distributions defined? I am reading a textbook on rigorous quantum mechanics and quantum field theory in which there often appears statements such as Let$A(\phi)$be a polynomial function defined on$\mathcal{S}'(\mathbb{...
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If $\omega$ is a closed $k-$current on the interior of $S_n:=\{x \in \Bbb{R}^n: 0 \leq x_i \leq 1, 0 \leq x_1+...+x_n \leq 1\}.$ Then $\omega = d \eta$ where $\eta$ is an extendible $(k-1)-$current or ...