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Results tagged with schwartz-distributions
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user 129831
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.
1
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Accepted
How to rigorously differentiate the convolution of a distribution and a $L^2$ function?
The last step is formally justified by 15.8, differentiation property of José Sebastião e Silva's "Integrals and orders of growth of distributions." (The paper is currently available here.)
More preci …
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3
votes
1
answer
508
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How to rigorously differentiate the convolution of a distribution and a $L^2$ function?
I want to prove the following: (Here, $W^{2,2}$ is a Sobolev space as defined in Evans, chapter 5; $S$ is a Schwartz space; and if $A$ is a distribution and $a$ a function, then $\langle A, a\rangle$ …
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1
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0
answers
202
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"Potential" for a divergence-free distribution
Edit: I would like to reopen this question since the linked potential duplicate question is not useful in showing that we get a tempered distribution as the potential, and I have found no easy way to …
- 1,326
4
votes
1
answer
469
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Uniqueness of distributional solutions to the Poisson equation
Let $S(\mathbb R^n)$ denote the space of all Schwartz functions on $\mathbb R^n$ equipped with the topology induced by the usual Schwartz semi-norms. Let $S(\mathbb R^n)^*$ denote its dual.
My questio …
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