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Results tagged with schwartz-distributions
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user 113756
A distribution is a continuous linear functional on the space $\mathcal{C}^{\infty}_c$ of smooth (indefinitely differentiable) functions with compact support. Though they appeared in formal computations in the physics and engineering literature in the late $19^{th}$ century, their formal setting was brought up by the work of S. Sobolev and L. Schwartz in the middle of the $20^{th}$ century.
3
votes
Integral representation of tempered distributions
No, and a simple example goes as follows:
$$
\begin{split}
K(\varphi_{1}\otimes\cdots\otimes\varphi_{N})
& =\left(\prod_{i=1}^{N}\prod_{k=1}^{n_i}\frac{\partial}{\partial x_k}\right)\varphi_1(0)\cdot …
- 6,357
2
votes
Accepted
Pointwise functional derivative as partial derivative
A functional derivative is what you need:
$$
\begin{split}
\frac{\partial f}{\partial \phi} &\triangleq \left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon} f(\phi+\varepsilon\varphi)\right|_{\varepsilon=0} …
- 6,357
4
votes
Prove that a given distribution is tempered
Edit: even if another answer has been accepted, I edited mine in order to correct (hopefully) the issues raised in the comments, and
possibly list more easily readable references to Łojasiewicz's sol …
- 6,357
2
votes
Generalizing a formula with distributions — Distributional Radon transform
Edit. While my first answer was more a suggestion on how to proceed, I decided to expand it in full. I also changed the former notation at some point in order to make it clearer, while still retaining …
- 6,357
5
votes
Accepted
Defining the value of a distribution at a point
The definition of the value of a distribution at a point you describe in your question does not seem flawed to me since, at least from the point of view of the independence on $\delta$-sequences, foll …
- 6,357
9
votes
Anti-delta function?
This is more a long comment following the ones of Mateusz Kwasnicki and Anixx and the nice answers given by Michael Hardy, Gro-Tsen, et al.
As stated by Mateusz, such a mathematical objiect cannot be …
- 6,357
7
votes
Accepted
Research topics in microlocal analysis
I can give a brief description of a (perhaps, currently the main) topic pertaining the application of microlocal analysis to the study of nonlinear PDEs, which seem the argument you are most intereste …
- 6,357