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6
votes
1answer
525 views

Distributions on product spaces

I hope this is suitable to MO. Question. Let $X$ and $Y$ be two open sets in $\mathbb R^n$ and $\mathbb R^m$, respectively. In what sense can we consider $\mathcal{D}^{\prime} \left(X\times Y\right)$ ...
7
votes
3answers
914 views

Rationale for Hadamard's finite part of a divergent integral

(Note: I asked this question a few days ago on math.stackexchange but didn't get any responses. I've therefore decided to post it here instead.) I have a problem justifying the throwing away the ...
3
votes
1answer
416 views

Ergodicity of Convoluted White Noise

I have a question regarding ergodicity in infinite dimensional spaces. Let $\mathcal{D}$ be the space of distributions on a Schwartz space, and let $\mu$ be the white noise process which exists by ...
1
vote
3answers
578 views

Solving $x\partial_x f = 0$ over distributions

Solving $x\partial_x f = 0$ over 'normal' functions is the same as solving $\partial_x f = 0$, i.e. one gets $f(x)=c_1$ as the complete answer. But over distributions (if my calculations are ...
1
vote
1answer
243 views

Division of distributions by polynomials.

I'm trying to solve the equation $(1-|x|^2)T = 0$, where $T$ is a tempered distribution. I know how to do this (it is a common exercise) in dimension $1$. How can I solve it in higher dimensions? ...
0
votes
2answers
606 views

Inverse Fourier transform of class of infinitely differentiable function with compact support

For which $f \in S(R^n)$, the Schwartz class, $\hat f \in D(R^n)$ ?
8
votes
2answers
323 views

Notion of generalized function/distribution for functional derivatives?

Is there any work on defining something analogous to a generalized function for functions whose domain is a Hilbert or Banach space? Is there an extension of the notion of Frechet/Hadamard/Gateaux ...
2
votes
1answer
2k views

Proving $\int_{0}^{\infty} \sin x/x dx=\pi/2$ by test functions and distributions

There are $C^{\infty}$ test functions in $L^{1}(0, \infty)$ that make the integral value of $\int_{0}^{\infty}(\sin x/x) \phi(x) dx$ range from 0 to $\infty$. What does narrowing these test functions ...
3
votes
1answer
536 views

An elementary introduction of Colombeau's generalized function theory

Hello, I am wondering whether anyone know an elementary reference for Colombeau's theory on the multiplication of distributions? I encountered the problem of the square of Delta function. I need a ...
2
votes
1answer
796 views

A good reference for the wave front set

Hello, I am wondering whether anyone know some good references for the theory of wave front set, microlocal analysis? I have some basic knowledge of distribution theory at the level of the Rudin's ...
1
vote
5answers
2k views

fourier transform of (real) exponential

Is it possible to make sense, in distributional sense, of the Fourier transform of the exponential function (defined over the whole real line)?
5
votes
3answers
618 views

Explicit isomorphism between distributions and universal enveloping algebra

The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is isomorphic to the algebra of distributions on the Lie group $G$ with support at the identity. The proof I have of this fact uses the ...
3
votes
1answer
1k views

Existence of weak limits

Background: Three months ago, I asked this question, which is a bit related to the following (if the answer to it was Yes, then answer to this one would be Yes too, but since that was a No, it still ...
5
votes
4answers
952 views

Distributions more complicated than the Dirac δ and derivatives

The responses to another question clarifies that the best known examples of distributions that are not measures, are the derivatives of the delta and such. What I want to know is: Is that the only way ...
11
votes
18answers
6k views

Good books on theory of distributions

Hi all. I'm looking for english books with a good coverage of distribution theory. I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions. Thanks in advance.
1
vote
2answers
396 views

What do we know about the space of finite order distributions ?

Hi, (Question updated) My question is about the space of distributions of finite order $\mathcal{D}'_F$ (say on $\mathbb{R}^n$). What do we know about it ? From in the information I gathered, it ...
22
votes
7answers
2k views

What examples of distributions should I keep in mind?

I'm learning a bit about the theory of distributions. What examples of distributions will help me develop good intuition? Definitions: Let $U$ be an open subset of $\mathbb{R}^n$. Write ...