The schwartz-distributions tag has no wiki summary.

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### 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 $\mathcal{D}^{\prime} \left(X\times Y\right)$ ...

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**3**answers

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### 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 ...

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**1**answer

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### 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 ...

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**3**answers

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### 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 ...

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**1**answer

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?
...

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**2**answers

606 views

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

For which $f \in S(R^n)$, the Schwartz class, $\hat f \in D(R^n)$ ?

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**2**answers

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### 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 ...

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**1**answer

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### 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 ...

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**1**answer

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### 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 ...

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**1**answer

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### 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 ...

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**5**answers

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### 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)?

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**3**answers

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### 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 ...

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**1**answer

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### 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 ...

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**4**answers

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 ...

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**18**answers

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### 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.

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**2**answers

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### 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 ...

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**7**answers

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### 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 ...