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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1 vote
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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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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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1 vote
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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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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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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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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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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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