# 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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### Making a Fourier Transform converge

Consider for example the following function \begin{equation} f(x)=(e^x+1)^{ik_1}\,. \end{equation} For $k_1 \in \mathbb{R}$, this is clearly not absolutely integrable and thus its Fourier transform ...
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### How do we derive the composed Dirac Delta function?

Suppose we have to compute $∂_{x}\theta(y-x)$, where $θ$ is the Heaviside step function, how do I find the result? Intuitively I would just use the chain rule $\partial_{x}\theta(y-x)=-\delta(y-x)$ ...
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1 vote
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### Laplacian of 1/r^n in the distributional sense

Let $\Omega = \mathbb{R}^3\setminus\{0\}$. Consider the function $$f_n \colon \Omega \to \mathbb{R},\quad \vec{x} \mapsto \frac{1}{\|\vec{x}\|^n}$$ with $n \in \mathbb{Z}^{+}$. I want to calculate ...
1 vote
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### Who is "the set of weakly differentiable functions"?

My question is pretty straightfoward: I would like to be able to think of "the set of weakly differentiable functions". Recall that: a function $f:I=(a,b) \rightarrow \mathbb{R}$ is weakly ...
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### Derivative of a distribution, including an integral of abs(x)

I'm currently reading a book on finite element analysis and I lack knowledge on higher level of mathematics. My question is regarding the derivative of a distribution. I'm stuck on an example ...
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### What is the limit for $x$ approaching zero of the dirac delta distribution $\delta(x)$?

I am trying to prove that $$\lim_{y\rightarrow 0} \delta(y-x) = \delta(x) .$$ To justify this problem, this comes from the orthogonality of the position eigenstates in quantum mechanics. Indeed, we ...
1 vote
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### Is the convolution of a tempered distribution and a Schwartz function also a function?

Let $T \in \mathscr{S}'(\mathbb{R}^n)$ and $f \in \mathscr{S}(\mathbb{R}^n)$. We may define their convolution as $$(T * f)(\varphi) = T(\tilde{f}*\varphi)$$ where $\tilde{f}(x) = f(-x)$. The above ...
1 vote
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### Evaluating $\int_{p} f(p) \delta(p+a) \Theta(p)= f(-a) \Theta(-a)$ for Heaviside theta function

I want to evaluate the following integral For given arbitrary function $f$ and Dirac Delta function $\delta$ with Heaviside Theta function $\Theta$, what is the form of \begin{align} \int_{p} f(p) \...
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### How can I get the properties of a function, which has an integral equal to zero?

Let $\varrho \in L^\infty_{loc}$, where the set of discontinuity points is a Lebesgue null set, and $\varrho * \varphi$ is a polynomial with maximum degree of $m$ $\forall \varphi \in C_c^\infty$. Now ...
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### How to prove$|\cdot|^{s}g\in\mathcal{S}'$? a question from Bahouri-Chemin-Danchin book "fourier analysis and nonlinear pde"

In page 27 and 28 of book "fourier analysis and nonlinear partial differential equations", proposition 1.36 the authors Bahouri-Chemin-Danchin give a proof of this proposition, but I really ...
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### Distributional Laplacian of Logarithmic function.

Here is the exercise: Compute the distributional Laplacian $\left(\text{in }\mathbb{R}^2\right)$ of $d(x,y)=\ln\left(\|(x,y)\|\right)=\ln\left(\sqrt{x^2+y^2}\right)$. Relate your answer to $\delta$ ...
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### Dividing a tempered distribution by a polynomial

Let $p=p(x_1,...,x_N)$ be a non-zero polynomial in $N$ variables (real coefficients). Let $\mathscr{S}$ be the Schwartz space on $\mathbb{R}^N$ and let $\mathscr{S}'$ be its topological dual (i.e. the ...
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### Sum in the distributional sense

Put $$S_n(\xi, x)=\sum_{k=0}^{+\infty} \frac{L_k^n(x)}{\xi-k}, x \in C, \xi \in C \backslash Z_{+}$$ Now, to compute the sum $S_n(\xi, x)$ we begin by writing it in terms of the Laguerre polynomial ...
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### Generalization of Dirac delta identity

The Dirac delta distribution obeys the following identity in $\mathbb{R}$ $$\lim_{\epsilon\to 0}\dfrac{1}{\pi}\dfrac{\epsilon}{\epsilon^2+x^2}=\delta(x)\tag1.$$ I know how to prove this using the ...
1 vote
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### From Trevés - Finite order distributions in terms of Radon measures

Trevés, Theorem 24.4, Topological Vector Spaces, Distributions and Kernels, Dover, p. 259. If $T$ is a distribution of finite order $\leq m$ in $\Omega$ with support $S \subset \Omega$, then given any ...
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### $d$-dimensional Fourier transform of $\cos|x|$

What is the $d$-dimensional Fourier transform of $\cos|x|$? More specifically, what does $$\int_{\mathbb{R}^d}\text{d}^dx\cos|x|e^{-i\vec{k}\cdot\vec{x}}\ ,$$ where $|x|=\sqrt{x_1^2+\dots+x_d^2}$, ...
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### Calculate the limit of $\langle \partial_x f,\rho_\varepsilon \ast \chi_B \rangle$

Let $f(x_1,x_2)=x_2^2\chi_E$ where $E=\{(x_1,x_2)\in \mathbb{R}^2:x_1\geq 0 \}$ and let $B$ be the unit ball in $\mathbb{R}^2$. Since $f$ is locally integrable, it can be interpreted as a distribution ...
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### Fourier integral representation of dirac delta function

Is the following proposition true? Proposition. For any $a,k\in \mathbb{R}$, \begin{equation} \int_a^{\infty} dx e^{ikx} = 2\pi \delta(k). \end{equation} (End) I think it is true based on the ...
1 vote
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### defintion of weak solution

i found that a weak solution of $-\Delta u=f$ where f $\in L^{2} (\Omega)$ is a solution u $\in H_{0}^{1}(\Omega)$ and it is solution in sense of distribution so what does it mean that that u is ...
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### Confused About Rudin's Opening Statements on his Chapter on Distributions

I have a few doubts regarding Rudin's opening paragraphs in Chapter 6 of Functional Analysis. The passage (with small edits) reads: A complex function $f$ is said to be locally integrable if $f$ is ...
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### Upper bound of sum of random variables

Suppose that we have random variables $X_1$, $X_2$ each drawn independently from Irwin Hall Distribution with same mean of 0 and different variances. If I have B1 and B2, which are high probability ...
1 vote
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### Does all the well defined product of Hyperfunction satisfy the product rule

I read in the first two chapters of Urs Graf's "Introduction to Hyperfunctions and Their Integral Transforms". On the page 114 of the book, The book well defined the product of ...
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### Is a function with good properties defined in a closed interval a generalized function?

I'm not a math student. I just want to touch distribution theory briefly. My question is, is a function with good properties defined in a closed interval a generalized function? For example, $f(x)=x$ ...
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### Confused about Georgiev's definition of $C^{\infty}_c$

I'm beginning to study Distributions, and I've encountered the following definition in Georgiev's Theory of Distributions: Such definition implies that $C^{\infty}_c$ (with the opology given by the ...
1 vote
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### What is the easiest way to define the canonical $LF$-topology?

There are many ways to define the canonical $LF$-topology on $C^{\infty}_c$, as explained here. Understanding that none of them is particularly easy, which method for defining the topology is the ...
I often see equation following: $$\int_{-\infty}^{\infty} \delta(x) dx=1 =\int_{-\infty}^{\infty} 1 \cdot \delta(x) dx$$ But, as i know, The distribution isn't possible for any function; it is only ...