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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7
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1answer
266 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 ...
2
votes
1answer
298 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
votes
1answer
228 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 ...
9
votes
1answer
364 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 ...
2
votes
0answers
319 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
598 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
votes
3answers
639 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 ...
8
votes
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 ...
8
votes
2answers
862 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
votes
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
votes
2answers
405 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 $\Omega=(0, \infty)$ ...
2
votes
2answers
193 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
votes
2answers
400 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
votes
1answer
374 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
votes
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
votes
1answer
186 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 ...
5
votes
2answers
765 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 ...
6
votes
2answers
666 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 ...
9
votes
2answers
221 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 ...
1
vote
0answers
327 views

Misapplication of the Hahn-Banach theorem

I was reading a comment on MathOverflow and did not understand what was being suggested. I also don't understand Anton's response: ...
7
votes
2answers
829 views

Dual of Sobolev space $W^{1,p}(U)$ for $U$ an arbitrary subset of $\mathbb R^n$

this question may be shameful, but nevertheless I can't help myself. Let $U \subset \mathbb R^n$ be arbitrary, in particular not the whole of the space itself. I wonder about the dual of the space ...
4
votes
1answer
338 views

Distributional derivative as linear approximation

Is it possible to characterize the distributional derivative as some sort of "best linear approximation" of a distribution (a la the Fréchet/Gâteaux derivatives), if viewed in the appropriate spaces? ...
6
votes
3answers
2k views

How to prove $\frac{d\theta}{dx} = \delta(x)$?

Here is a problem from Griffith's book Introduction to E&M. Let $\theta(x)$ be the step function $$\theta = \begin{cases} 0, & x \le 0, \\ 1, & x \gt 0. \end{cases} $$ The ...
4
votes
4answers
1k views

Distribution theory book

I'm looking for a good book on distribution theory (in the Schwartz sense), I have the basic knowledge as given in Grafakos' Classical Fourier Analysis, but I want to know more about it. Is the ...
8
votes
3answers
6k views

what is product of delta function with itself?

what is the product of delta function with itself ? what is the dot product with itself ?
4
votes
2answers
323 views

Operators commuting with translations

Let $T$ be a bounded linear operator on $L^2(\mathbb R)$. So, let us now assume that $T$ commutes with the translations $\tau_x$. How do I now show that $T$ is given by a convolution with respect to a ...
4
votes
3answers
531 views

Derivatives distribution

Let $f$ be a distribution on $\mathbf{R}^n$ (in the Schwartz sense) such that $$\frac{\partial f}{\partial x_i} = 0 \text{ for $i = 1, \ldots, n$.}$$ Then how to prove that $f$ is a constant? I had ...