Use this tag for questions about Schwartz distributions, also known as Generalised 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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$C^{\infty}$ function represented by the diverging integral

There is a theorem (see Treves: "Introduction to Pseudodifferential and Fourier integral operators") that states that the kernel of any pseudodifferential operator, i.e. the distribution $$ K(x,y) ...
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0answers
167 views

What does it mean to carry out calculations modulo $S^{- \infty}$

I have trouble making sense of the following remark that is given in a set of lecture notes I am currently going through on my own: " Oscillatory integrals are used for the study of singularities of ...
7
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1answer
669 views

Meaning of polynomially bounded

I am currently reading through some lecture notes on Fourier Transform and Distributions on my own, and came upon the notion of a polynomially bounded function. I am not sure I understand this ...
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1answer
392 views

how could i prove this (delta function representation)

how could i prove that the sucession $ \frac{k^{k+1}}{k!}t^{k}e^{-kt} $ tends in the limit $ k \to \infty$ to the delta function $ \delta (t-1) $ this is used inside the post 's inversion formula ...
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2answers
140 views

Is the following derivative defined?

I am new to this site so I am not sure if this is the right place to be asking this question but I will try anyway. I am reading an economics paper for my research and the author does the following: ...
2
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2answers
200 views

Simplifying the generalized function $x^{\lambda}_+$ in the strip $-n - 1 < \mbox{Re}\lambda < -n$

Note: this post is a follow up to an earlier question. The (divergent) integral of $x^{\lambda}_+$ can be analytically continued into the region Re $\lambda > -n - 1$, $\lambda \ne -1, -2 , \ldots ...
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4answers
3k views

Which of these two ways to take the derivative of a delta function times another function is correct?

A well known identity of the Dirac delta function is that for any function $f(x)$: $$ \delta(x) f(x) = \delta(x) f(0). $$ If we take the derivative of the right hand side we get: $$ ...
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2answers
180 views

Tempered distributions of finite order?

Is every tempered distribution of finite order? It seems that yes with the definition.
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4answers
251 views

Borel Measure such that integrating a polynomial yields the derivative at a point

Does there exist a signed regular Borel measure such that $$ \int_0^1 p(x) d\mu(x) = p'(0) $$ for all polynomials of at most degree $N$ for some fixed $N$. This seems similar to a Dirac measure ...
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1answer
185 views

Showing that $\int_0^1 x^{\lambda} [ \: \phi(x) - \phi(0)\: ] dx$ is convergent for $\lambda > -2$

Id' appreciate help understanding why the integral $$ \int_0^1 x^{\lambda} [ \: \phi(x) - \phi(0)\: ] dx $$ is convergent provided $\lambda > -2$, where $\phi \in \mathcal{D}(\mathbb{R})$. To ...
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1answer
460 views

Sequential and topological duals of test function spaces

Given a test function space, in particular $\mathcal{S}=\mathcal{S}(\mathbb{R}^n)$ (the Schwartz space) or $\mathcal{D}=\mathcal{D}(\mathbb{R}^n)$ (the space of compactly supported smooth test ...
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1answer
295 views

Delta function representation

Suppose I have a set of functions $(f_\epsilon)$ such that as $\epsilon\to 0$, $f_\epsilon\to F$ s.t. $F(x)=0$ for $x\neq 0$ and $F(x)=\infty$ for $x=0$; $\int_{-\infty}^\infty f_\epsilon(x) dx=1$ ...
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2answers
350 views

When is the convolution with a tempered distribution again a tempered distribution?

If $f$ is a Schwartz function on $\mathbb R^n$ and $g \in L^1(\mathbb R^n)$, then if $g$ is the Poisson kernel, is $f\ast g$ a Schwartz function? are there any known sufficient conditions on $g$ to ...
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0answers
81 views

Deduce the global differential equation from the pointwisely defined equation in Fourier space

Let $G\in \mathcal{F}(\mathbb{R}^{n+1})'$ be a distribution on the space of spatial Fourier transform'able function, ie an $L^1_{\mathrm{loc}}(\mathbb{R^{n+1}})$ function, $G = G(t,\xi)$. Assume ...
9
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1answer
173 views

Oscillatory integral giving me the willies

So now that my term's over, I've been brushing up on my quantum field theory, and I came across the following line in my textbook without any justification: ...
0
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1answer
244 views

An (elementary?) Question on Weak Derivatives

Let $U\subset \mathbb{R}^{n}$ be an open set and $f:U \to \mathbb{R}$ a continuous function which is piecewise $C^{1}$. This is: there is a partition of $U$ by (say, a finite number of) open sets ...
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2answers
580 views

How do I write the 2D Dirac delta in a manifestly rotationally invariant form?

Consider the following integral over a 2D plane, $$\iint \mathrm{d}^2\mathbf{k}\ e^{i\mathbf{k}\cdot\mathbf{r}} = 4\pi^2\delta^2(\mathbf{r})$$ This is a Fourier transform of a distribution which is ...
0
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1answer
140 views

The convergence of distribution

Let $\mathcal S'(\mathbb R^n)$ the space of all continuous linear functions from the Schwartz space $\mathcal S(\mathbb R^n)$ to $\mathbb C$, and $\mathcal D'(\mathbb R^n)$ the space of all continuous ...
5
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1answer
298 views

Laplace transform and differentiation

Let $F(s)$ be the Laplace transform of $f(t)$: $$F\left(s\right)=\int_{0}^{\infty}e^{-st}f\left(t\right)dt$$ It then follows that $f(t)$ can be recovered from $F(s)$ by the inverse Laplace ...
3
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1answer
738 views

The constant distribution

If $u$ is a distribution in open set $\Omega\subset \mathbb R^n$ such that ${\partial ^i}u = 0$ for all $i=1,2,\ldots,n$. Then is it necessarily that $u$ is a constant function?
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1answer
76 views

Infinite sum of rapidly decreasing test functions

Let $\phi$ be a rapidly decreasing test function on $\mathbf R$ and define $\tau_n$ so that $\tau_n\phi(x)=\phi(x+n)$. I would like to show that the series $\begin{align}\sum_{n \in \mathbf Z} \tau_n ...
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3answers
254 views

Strange functional equation ( hyperfunctions? )

Can we solve this strange functional equation? $$ f(x+i\epsilon)-f(x-i\epsilon) = g(x) $$ I believe that the solution is the Hilbert (finite part) transform of the function g(x) however I do not ...
3
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1answer
640 views

weak convergence to delta function

Let $f_\epsilon\in L^1(\mathbb{R}^n)$ be a function which depends on a parameter $\epsilon\in(0,1)$, and is such that $\operatorname{supp}{f_\epsilon}\subset\{|x|\leq\epsilon\}$, the total integral ...
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2answers
4k views

Proof of Dirac Delta's sifting property

A common way to characterize the dirac delta function $\delta$ is by the following two properties: $$1)\ \delta(x) = 0\ \ \text{for}\ \ x \neq 0$$ $$2)\ \int_{-\infty}^{\infty}\delta(x)\ dx = 1$$ I ...
3
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1answer
284 views

wavefront sets of distributions

I'm trying to understand why the wavefront set of the distribution $1/(x+i0)$ is given by $\left\{(x,\xi):\xi=0\ \textrm{or}\ x=0,\xi\geq0\right\}$, and why that of $1/(x-i0)$ is ...
3
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1answer
314 views

The limit of locally integrable functions

If ${f_i} \in L_{\rm loc}^1(\Omega )$ with $\Omega $ an open set in ${\mathbb R^n}$ , and ${f_i}$ are uniformly bounded in ${L^1}$ for every compact set, is it necessarily true that there is a ...
2
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2answers
139 views

The limit in the distribution

Let for $j\in\mathbb N$, $f_j \in \mathcal D'(R^n)$ (distribution) and $f_j$ is locally integrable for all $j$. If $f_j \to f$ in the sense of distribution, is it necessarily true that $f$ is ...
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1answer
570 views

Tensor products of functions generate dense subspace?

Let $X$ and $Y$ be two spaces in certain category, $F(\cdot)$ a functor associating each space with a function space (with certain topology). Assume that for any $f\in F(X)$ and $g\in F(Y)$, $f\otimes ...
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3answers
2k views

Property of Dirac delta function in $\mathbb{R}^n$

How does one prove the following identity? $$\int _Vf(\pmb{r})\delta (g(\pmb{r}))d\pmb{r}=\int _S\frac{f(\pmb{r})}{|\text{grad} g(\pmb{r})|}d\sigma$$ where $S$ is the surface inside $V$ where ...
4
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1answer
528 views

understanding of the classical definition of Green's function

I learn the classical definition of Green's function from Hunter's Applied Analysis. Consider the second-order ordinary differential operators $A$ of the form $$Au=au''+bu'+cu,$$ where $a,b$ and $c$ ...
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1answer
349 views

How to calculate inverse c.d.f. for Zipf distribution

How can i calculate the Inverse c.d.f. for Zipf distribution? This distribution has: $$ \mathrm{pdf}(x) = \frac{1}{x^s \times H_{N,s}} $$ $$ \mathrm{cdf}(x) = \frac{H_{x,s}}{H_{N,s}} $$ where $s ...
7
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1answer
255 views

Questions about Fubini's theorem

I learned the following from Hunter's Applied Analysis. Denote the Schwartz space $${\mathcal S}({\mathbb R}^n):=\{\varphi\in C^{\infty}({\mathbb R}^n):\sup_{x\in{\mathbb ...
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1answer
283 views

Why 2 distributions can not be multiplied? [duplicate]

Possible Duplicate: what is product of delta function with itself ? why $2$ or more dirac delta distributions can not be multiplied ?? i mean to define a coherent product of $d(x)$ x ...
6
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1answer
220 views

Difficulties in solving a PDE problem

This is an exercise in "Variation et optimisation des formes", chapter 3, Ex. 3.8. The preliminaries are: $$D=(0,1)^2,\ f \in L^2(D),\ x_{ij}=(i/n, j/n),\ 0<i,j<n,$$ $$\Omega_n = D\setminus ...
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1answer
323 views

Origin of the name 'test functions'

This is a very simple question really: where did the name 'test functions', used nowadays when speaking of infinitely differentiable and compactly supported functions, come from? More to the point: is ...
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0answers
305 views

Seeking rationale for Hadamard's finite part of a divergent integral

I have a problem justifying the throwing away the divergent term in order to obtain Hadamard's finite part. I find this step to be highly unusual and it is not obvious to me how the resulting ...
4
votes
1answer
578 views

The (distributional) Fourier transform of the unit step function

I want to calculate the Fourier transform of the unit step function (given by $\phi(x) = 1$ for $x \geq 0$ and $\phi(x) = 0$ for $x < 0$) regarded as a tempered distribution. Note that I don't ...
3
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3answers
583 views

Confusion on unit impulse function $\delta(t)$

$\delta(t)$ is a singular function and when I'm learning Signals and Systems I learned that $\delta(t)$ is an even function, and all of its odd order derivatives are odd function. Then we have ...
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2answers
1k views

When can one expect a classical solution of a PDE?

When solving a PDE, there may be a classical solution or a weak solution (or distribution solution). But I am wondering that when people talk about "finding a solution" to some PDE, what do they refer ...
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2answers
778 views

How to prove that the Cantor ternary function is not weakly differentiable?

I am using the standard cantor ternary function $f$ here, as cited in this Wikipedia page. It is an example of continuous, monotone increasing, but not strictly monotone increasing function with zero ...
8
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1answer
273 views

What is the name for the archetypical example of a test function, $\varphi(x)=e^{1/(x^2-1)}$?

$$ \varphi(x)=e^{1/(x^2-1)} $$ This function (on the interval $\quad]\!-1,1[ \,\,\, $, outside of it simply $\equiv0$) is used as the typical example of a test function / bump function, I have so ...
2
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2answers
372 views

Distribution of infinite order

This is an exercise from Functional Analysis By Walter Rudin on page 178 from chapter Test functions and Distribution. I am having trouble arguing for 2nd part. Question: For $\Omega=(0, \infty)$ ...
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2answers
179 views

Characterization of delta Distribution

I encountered this problem while solving a problem related to characterization of delta function upto constant multiple. $\phi \in D(R)$, Space of compactly supported infinitely differentiable ...
2
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2answers
383 views

Delta function in curvilinear coordinates

I have been looking everywhere but I am unable to prove $$\delta(\vec{x}-\vec{a}) = \frac{1}{fgh}\delta(x_u-a_u)\delta(x_v-a_v) \delta(x_w-a_w)$$ Where $f,g,h$ are scale factors for an orthogonal ...
0
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1answer
361 views

Proving that the bi-laplacian of a radial basis function is the dirac delta

According to equation (2.14) of the paper "The Uniform Convergence of Thin Plate Spline Interpolation in Two Dimensions" a radial basis function $\phi(\parallel x \parallel)$ has the property $$ ...
12
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3answers
1k views

Delta function integrated from zero

I am trying to understand the motivation behind the following identity stated in Bracewell's book on Fourier transforms: $$\delta^{(2)}(x,y)=\frac{\delta(r)}{\pi r},$$ where $\delta^{(2)}$ is a ...
5
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1answer
178 views

Tempered distribution concentrated in a lower dimensional manifold

Question: What can you conclude about a tempered distribution $G\ \in\ S'(R^n)$ that is concentrated in some k-dimensional manifold $M\ \subset\ R^n$ (for k < n)? More specifically, is there a ...
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2answers
682 views

On distributions over $\mathbb R$ whose derivatives vanishes

Let $I \subset \mathbb R$ be open, $u \in \mathcal D'(I)$ be a distribution whose distributional derivatives vanishes (i.e. is zero for all test functions, which we may assume to be complex valued ...
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2answers
597 views

Topologies on the space $\mathcal D'(U)$ of distributions

In my analysis lecture I am given a topology on the space of distributions as follows: Let $u_k$ be a sequence in $\mathcal D'(u)$, $u \in \mathcal D'(u)$. We say $u_k \rightarrow u$, if ...
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2answers
215 views

How do different notions of “distribution” relate to one another?

In reading "Real Analysis: Modern Techniques and Their Applications" (Folland), I've come across a few different notions of "distribution" or "distribution functions." The distribution function of a ...