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).

Distributions are object that generalize the notion of function. They are linear functionals on a set of test functions into the real numbers. The set of test functions is usually $\mathcal{D}(\mathbb R^n)=\mathcal{C}^{\infty}_c(\mathbb R^n)$. The basic idea is to to reinterpret functions as linear functionals acting on a space of test functions.

If we use a larger test space, such as $\mathcal{S}(\mathbb R^n)$ we obtain a smaller space of distributions, called tempered distributions. The space of distributions is usually denoted by $\mathcal{D}'(\mathbb R^n)$, while tempered distributions are usually denoted by $\mathcal{S}'(\mathbb R^n)$.

Distributions are heavily used in partial differential equations (when classical solutions don't exist there might still be distributional solutions), physics and engineering.

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Are weak derivatives and distributional derivatives different?

Given a real function $f\in L^1_{\text{loc}}(\Omega)$, we define both weak or distributional derivatives by $\int f'\phi = - \int f \phi'$ for all test functions $\phi$. Now, take $\Omega = (-1,1)$, and $f(x) = I_{x>0}$, an indicator function.…
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Is the Dirac Delta "Function" really a function?

I am given to understand that the Dirac delta function is strictly not a function in the conventional sense and it is a "functional or a distribution". The part which I can not understand why the Delta "function" makes sense only when it acts on…
noir1993
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Laplacians and Dirac delta functions

It is often quoted in physics textbooks for finding the electric potential using Green's function that $$\nabla ^2 \left(\frac{1}{r}\right)=-4\pi\delta^3({\bf r}),$$ or more generally $$\nabla ^2 \left(\frac{1}{|| \vec x - \vec…
Jenna
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Sobolev space $H^s(\mathbb{R}^n)$ is an algebra with $2s>n$

How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have $\lVert uv\rVert_s \leq C \lVert u\rVert_s \lVert…
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How to prove that the derivative of Heaviside's unit step function is the Dirac delta?

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 question is how to prove $\frac{d\theta}{dx} = \delta(x)$. I think since…
Jichao
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Two possible definitions of "vector-valued distribution"

Let $X$ be a reflexive real Banach space, the complex case should be totally analogous. Define $$\tag{1} \mathcal{D}^\star(0, T; X)=\left\{ u\colon \mathcal{D}(0, T)\to X\ \text{linear and continuous}\right\} $$ where the topology on…
Giuseppe Negro
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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 $g\in F(Y)$, $f\otimes g := f(x)g(y)\in F(X\times Y)$. Denote by…
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Derivative of a Delta function

I know questions similar to this one have been asked, but there is a particular aspect that I'm confused about that wasn't addressed in the answers to the other ones. I'm dealing with an expression which I have simplified into something…
TeeJay
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What does "test function" mean?

I am trying to learn weak derivatives. In that, we call $\mathbb{C}^{\infty}_{c}$ functions as test functions and we use these functions in weak derivatives. I want to understand why these are called test functions and why the functions with these…
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Dirac delta in polar coordinates

Given $$x=r\,\cos\theta\\y=r\,\sin\theta$$ and $$x'=r'\,\cos\theta'\\y'=r'\,\sin\theta'$$ how can I express $$\delta(x'-x)\delta(y'-y)$$ in terms of the polar coordinates? And the more general case: $$\delta(x'-x-a)\delta(y'-y-b)$$
JFNJr
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Various kinds of derivatives

Let $f\colon \mathbb{R}\to \mathbb{R}$ be a measurable function. Let us introduce the following notions of "derivative" of $f$. Classical derivative. The unique function $f'_c$ defined pointwise by the following:$$\lim_{h\to 0} \frac{…
Giuseppe Negro
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Which functions are tempered distributions?

Today's problem originates in this conversation with Willie Wong about the Fourier transform of a Gaussian function $$g_{\sigma}(x)=e^{-\sigma \lvert x \rvert^2},\quad x \in \mathbb{R}^n;$$ where $\sigma$ is a complex parameter. When $\Re (\sigma)…
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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 2-dimensional delta function. Starting with something…
Leo Alekseyev
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If a Radon measure is a tempered distribution, does it integrate all Schwartz functions?

The question might at first sight sound like the answer is trivially "yes", so let me clarify the question a bit. Consider given a nonnegative Radon measure $\mu$ on $\mathbb{R}^n$. Let $\mathcal{D}(\mathbb{R}^n)$ denote the space of real test…
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Distributions on manifolds

Wikipedia entry on distributions contains a seemingly innocent sentence With minor modifications, one can also define complex-valued distributions, and one can replace $\mathbb{R}^n$ by any (paracompact) smooth manifold. without any reference…
The Vee
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