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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Is fourier series necessarily obey the convergence for an arbitrary operator?

In the traditional real analysis, the convergence of the fourier series worked for the most of the functions such as the one's in $L^2(P)$ or $C^1(T)$. Which was quite useful in the field of the ...
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Solving the convolution equation $U*g=\sin{2x}$ where $g(x)=e^{-|x|}$.

The problem is as stated in the title but in more detail: find all tempered distributions $U\in\mathcal{S'(\mathbb{R})}$ that solve the convolution equation given in the title. My approach uses the ...
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What is the distribution of the elements of the Moore-Penrose inverse?

Assuming $A$ is an $m \times n$ matrix (with $n \ge m$) of normally distributed elements with $\mu_A = 0$ and $\sigma_A = 1$, is there a mathematical formulation for the distribution of the elements ...
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Why is $\int_{-\infty}^\infty x^{-2}\delta(1/x) dx=1$?

Following the answer by @Carlo Beenakker here, $\int_{-\infty}^\infty x^{-2}\delta(1/x) dx=1$, but I failed to understand, why. Can anyone please explain it in simple terms?
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How can you derive the spacetime Fourier transform of the free Schrodinger evolution rigorously?

I'm trying to compute the spacetime Fourier transform of the free Schrodinger evolution. Consider $f\in L^2(\mathbb{R}^d)$ and $e^{it\Delta}f=:\mathcal{F}^{-1}(e^{-it|\xi|^2}\hat{f}(\xi))$ its free ...
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Is there any discrete equivalent to Gaussian mixture model?

I have studied Gaussian mixture model and other mixture models like Bernoulli mixture model etc. I want to know is there a mixture model with appropriate distribution that can represent the softmax ...
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Distributional Solutions to $xu'=0$

I want to solve the differential equation $xu' = 0$ for some distribution $u \in \mathcal{D}(\mathbb{R}^n)$. It seems apparent that the solution $u$ should be of the form $u = c_1 + c_2H$ where $c_1,...
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$T*\varphi=\sin(x)$

I am looking for a distribution $T:\mathcal D(\mathbb R)\to\mathbb C$ and a test function $\varphi\in C_c^{\infty}(\mathbb R)$ with $$T*\varphi = sin(x),$$ but I can't think of any pair $(T,\varphi)$ ...
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Integral of function with any test function along curve vanishes. Does that imply that the function must vanish?

Situation: I have a smooth curve $\gamma:(a,b)\to\mathbb R^n$ and a locally integrable function $f:(a,b)\to\mathbb R$ for which I know that $\int_a^b f(t)\phi(\gamma(t))\text{d}t = 0$ for any smooth, ...
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Find estimates for $\alpha_0$ and $\alpha_1$ and covariancematrix

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $$f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$$ and we ...
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Integrating distributions over submanifolds

Distributions may be "integrated against" (i.e. evaluated on) test functions, the following notation (for $T$ a distribution) is fairly common: $$ T(f) = \int T(x) f(x)\, \mathrm{d}x $$ I'm ...
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Change of variables in $W(0,T)$

Consider $W(0,T) = \{u \in L^2(0,T,V) \text{ with }u_t \in L^2(0,T,V^*)\}$. Pick as $V$ any Hilbert space $H^1_0\subseteq V \subseteq H^1$. One can therefore write the integral $\int_\Omega u_t(t)v(t)...
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Verify Fundamental Solution of the 3 Dimensional Laplace Operator

I would like to verify that $\frac{1}{4\pi |x|}$ is the fundamental solution of the 3 dimensional laplace operator so that $$ \triangle \frac{1}{4 \pi |x|} = \delta(x) $$ What I have tried: I think ...
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Proof of $\lim\limits_{A\to \infty } \int_{-A}^{A} \frac{\sin (\alpha x)}{x} f(x) dx = \pi f(0)$

Proof of the equation $$\lim\limits_{A\to \infty } \int_{-A}^{A} \frac{\sin (\alpha x)}{x} f(x) dx = \pi f(0)$$ Preface This is solely for context A distribution is a linear operator (a functional) ...
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Integrating Dirac delta distribution from $0$ to $1$.

Let $\delta$ be rigorously defined as a generalized function (lim of a function). I am guessing that $\int_{-1}^0\delta(x)d x=\int_0^1\delta(x)d x=\frac{1}{2}$? Also, let $E$ denote a set contains 1/3 ...
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Infinite-countable state space Markov Chain

I tried to review a few posts related to the infinite-countable state space Markov chain and its stationary distribution. However, I could not solve the problem myself. It relates to my previous post ...
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Independent and identically distributed random variables example

My question: Does such an example exist? Let $X_1$,$X_2$ be independent and identically distributed random variables, there exist $w \in \Omega$ such that $X_1(w) \neq X_2(w)$ My try: It's easy if ...
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How to find the fundamental solution of the operator$I-\Delta$?

I want to find a distribution $E$ which satisfies $(I - \Delta) E = \delta$. I tried the usual method of finding the fundamental solution of Laplace operator. Using polar coordinate and calculating $ \...
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Convergence of series in $\mathcal{S}'(\mathbb{R})$ implies boundedness of coefficients

Let $f(x)=\sum_{n=-\infty}^{\infty} c_ne^{inx}$ in $\mathcal{S}'(\mathbb{R})$. To show: $|c_n|\le M|n|^m$ for $|n|\ge 1$, some $M>0$ and $m\in \mathbb{N}$. We have $\forall \phi\in \mathcal{S}(\...
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Derivative in the distributional sense

Let $f\in L_{loc}(\mathbb{R})$, we define the function $g(x)=\int_0^xf(t)dt$. The quesion is to show that the derivative of $g$ is $f$ in the in the distributional sense. I know the locally integrable ...
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Prove that the following exmaple is a distribution

For $a>0$ I have to show that $$ \langle f_a,\phi \rangle = \int_{-\infty}^a + \int_a^\infty \frac{\phi(x)}{|x|}dx + \int_{-a}^a \frac{\phi(x) - \phi(0)}{|x|}dx $$ is a distribution. I don't ...
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5 votes
2 answers
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How to make $\int_0^\infty \delta(x) dx = \frac12 $ rigorous using generalized functions?

There are many versions of the theory on generalized functions. The most famous one is the distribution theory of Schwartz, where test functions are smooth and compactly supported. In the Schwartz's ...
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Deriving the mean of truncated $t$ distribution

I am trying to derive the mean of right truncated $t$-distribution but cannot solve the integration further. I tried to find the truncated mean as follow: The pdf of right truncated distribution is; $\...
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Problems with integral substitution that involves weird physics index notation and distributions.

I‘m supposed to show that the 3x3 matrix T, whose components are given by: $$ T^{ij}=\int_{\partial B_1(0)} \frac{x^i x^j}{|x||x|}d \Omega $$ ($x^i$ the i’th component of $x$, $\partial B_1(0)$ the ...
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Why is the integral from $0$ to $\infty$ of $(\phi(x)-\phi(-x))/x$ well defined if $\phi$ is a smooth function with compact support?

I am trying to understand the proof that the principal value of 1/x is a distribution : Ok so I understand that we decomposed $\phi$ into its even and odd part and the odd part disappeared so we are ...
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2 answers
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Applying Fourier transforms in two variables separately

Let $G(x, t) := (4\pi t)^{-d / 2}e^{-|x|^2 / 4t}$ for all $x \in \mathbb{R}^d$ and $t \in (0, \infty)$. The function $G$ is not integrable on $\mathbb{R}^d \times (0, \infty)$ since $$ \int_0^\infty \...
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Ambiguous answer for simple integral involving Dirac delta function

Are both of these answers to the following integral correct: \begin{equation} \int_0^1\int_0^1dx_1 dx_2\delta(1-x_1-x_2)\overset{?}{=} 1\text{ or }\sqrt{2} \end{equation} I can "justify" ...
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Uniform convergence of of series of functions

Let $p\in \mathbb{N}$ and $(a_n)_{n\in \mathbb{Z}}$ a sequence of complexe numbers such that $$|a_n|\le (1+|n|^p)\quad \quad \forall n\in \mathbb{Z}$$ Could you please help me to show that $$\sum_{n\...
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Solution in the distribution set

Could you please help me to prove that the solution of $$(1-\exp^{2i\pi x})T=0$$ in $D'(\mathbb{R})$ is $\sum_{n\in \mathbb{Z}}c_n\delta_n$ where $c_n=C$ for all $n\in \mathbb{Z}$
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2 votes
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If $\psi(t)=\displaystyle\int\limits_{-\infty}^{t}\varphi(x)dx$ then $\psi\in\mathbb{S}(\mathbb{R}) $

Exercices : Let $\varphi\in\mathbb{S}(\mathbb{R}) $ such that : $\displaystyle\int\limits_\mathbb{R}\varphi (x)dx=0 $ Let defined : $$\psi(t)=\displaystyle\int\limits_{-\infty}^{t}\varphi(x)dx$$ Then ...
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$P[X_n \in [\alpha, \beta]]\geq 1-\epsilon$ for all $n \in \mathbb{N}$.

Let $(X_n)$ r.v. such that converge in distribution to $X$. (a) Prove that for each $\epsilon > 0$ exists $\alpha,\beta \in \mathbb{R}, \alpha <\beta$, such that $$P[X_n \in [\alpha, \beta]]\...
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2 votes
1 answer
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What does it mean for a distribution to fail to be a smooth function?

I am trying to understand what is meant by a singular support of a distribution: So singular support is defined as the complement of the largest open set on which T (the distribution) fails to be a ...
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Question about this definition of support of a distribution

I am just starting to learn about distributions and I have a question about the implications of this definition: I don't understand why $U_x$ has to be contained in $V$ here. I believe the definition ...
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1 answer
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In what sense are usual probability distributions like normal, binomial, etc distributions in the formal definition?

I just learned the formal definition of a distribution: a linear continuous functional $T$ from the vector space of test functions $C_{c}^{\infty}$ to its field of scalars $\mathbb{K}$. Ok, I guess. ...
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1 answer
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Covariance of two additions of random variables

I have a set of independent random variables $A_{1}, A_{2}, ... , A_{n}, n \geq 7$ with $ \forall i \in[1, n], \ \mathbb{E}(A_{i}) = 0 \ \& \text{ Var}(A_{i}) = 1$ I have two other random ...
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How to derive the von Mises-Fisher distribution by restricting an isotropic normal distribution to a the unit sphere?

I've heard that one way of characterizing the von Mises-Fisher distribution is to restrict an isotropic normal distribution to a unit sphere. How is this done in practice? I know that the density ...
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1 vote
0 answers
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Fourier analysis with Lipschitz condition

Suppose $u$ is a distribution in $\mathbb{R}$ , with compact support $K$ and whose Fourier transform $\hat{u}$ is a bounded function in $\mathbb{R}$ . Let $f\in L^1(\mathbb{R})$ and $\hat{f}=0$ on $K$ ...
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1 vote
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Solving Backward Heat Equation with a Backward Heat Kernel?

Let $D>0$ be a constant. Imagine we have the following forward heat conduction problem: \begin{align*} \begin{cases} \partial_t u = D \partial_x^2u &, \quad (x,t) \in \mathbb{R} \times (0, \...
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1 vote
1 answer
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Prove that $1/(1+|x|^2)^n$ is integrable and find its value

I was studying the dual space $\mathscr{S}'(\mathbb{R}^n)$ of Schwarz space (tempered distributions). I was given a proof of the claim that $L^p(\mathbb{R^n})\subset \mathscr{S}'(\mathbb{R}^n)$, which ...
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2 votes
1 answer
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Fourier transform of $\frac{(x-iw)^\alpha}{(x+iw)^\alpha}$

Let $w>0$ and $\alpha>0$. I want to compute the Fourier transform of $\frac{(x-iw)^\alpha}{(x+iw)^\alpha}$ in the distribution sense, i.e., evaluate $$\int_{-\infty}^\infty \frac{(x-iw)^\alpha}{(...
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1 vote
1 answer
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Circumference of a circle using integral representation of Dirac delta

I am trying to obtain the circumference of a circle of radius $r$ in a rather complex way because I am practising with the integral representation of the Dirac delta. Let me refer to the circle as $C$ ...
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5 votes
3 answers
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How does the divergent sum $\sum_{n=1}^\infty\cos(2n\gamma)\sin(2nt)$ correctly evaluate an integral? Surely distributions don’t apply here

$\newcommand{\d}{\,\mathrm{d}}\newcommand{\res}{\operatorname{Res}}$Note: I don’t know any distribution theory myself, but I was informed by someone else and hinted to by this answer that my problem ...
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1 answer
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Fourier transform of $e^{-x} H(x)$

Find the Fourier Transform of $e^{-x}H(x)$. I can find the answer using general definition of Fourier transform. But I find it in the exercise Distribution chapter. So, I want to do it using ...
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Derivative of P.V. $(\frac{1}{x})$ in the distribution sense [duplicate]

Show that the derivative of P.V. $(\frac{1}{x})$ is $f^\prime[\delta] = - \ \text{lim}_{\epsilon \rightarrow 0} \int_{|x|\geq \epsilon} \frac{\phi(x) - \phi(0)}{x^2} dx$ To prove this I have written $\...
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1 answer
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Prove that $f$ is a linear combination of $\delta^{(k)}$ for $k=0, 1\ldots, n-1$. [duplicate]

Let $f$ is a tempered distribution such that $x^nf =0$ for an integer $n$. Prove that $f$ is a linear combination of $\delta^{(k)}$ for $k=0, 1\ldots, n-1$. $\delta$ is Dirac delta function. Please ...
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Fourier Transform of $(2H(x)-1)\cdot\delta(2H(x)-2)$

I am trying to calculate the Fourier transform of of the function (distribution): \begin{cases} \delta(x-1) & x>0, \\ -\delta(-x-1) & x<0. \end{cases} I tried ...
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  • 133
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Solutions $\lambda$ of $\displaystyle 1-\lambda \mathcal{F}(e^{-ix})(\xi)=0$

I am looking for reals $\lambda$ such that $$1-\lambda \mathcal{F}(e^{-ix})(\xi)=0,$$ where $\mathcal{F}$ is the Fourier transform. Using the Dirac distribution $\delta$, I found that the Fourier ...
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2 votes
1 answer
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Space of test functions $\mathcal{D}$

In $\mathbb{R}$, let $\psi(x)=e^{-1/x^2}$ for $x>0$ and zero otherwise. If $\mathcal{D}$ is the class of test functions, $\phi\in \mathcal{D}(a<x<b)$ for $\phi = \psi(x)\psi(1-x)$, provided $...
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4 votes
0 answers
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Heat equation with time-varying Neumann condition

Suppose that $u$ is the solution to the heat equation with mixed Neumann and Robin boundary conditions \begin{align} &\partial_tu(t,x) = k \partial_{xx} u(t,x), &&t>0, 0<x<L, \\ &...
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  • 1,158
0 votes
1 answer
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Integral equation $f_i(x) = \sum_j (u_{ij} \ast f_j)(x)$, where $u_{ij}$ are tempered distributions, using FT

Consider the following "generalization" to the integral equations, where I am interested in solving for functions $f_1, f_2, ..., f_n$, and $u_{ij}$ are some known tempered distributions. $$ ...
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