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Results tagged with schwartz-distributions
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user 39261
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.
7
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0
answers
142
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Characterization of tempered distributions from tempered sequences
Let $\mathcal{D}(\mathbb{R})$ be the space of compactly supported and infinitely smooth functions for its usual topology. Let
$\mathcal{D}'(\mathbb{R})$ be the topological dual of $\mathcal{D}(\mathb …
- 2,306
5
votes
1
answer
170
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Critical Smoothness on Besov Spaces $B^s_{p}$: how does it evolved with $p$?
We denote by $B_{p}^s(\mathbb{T}) := B_{p,p}^s(\mathbb{T})$ the Besov space over the circle $\mathbb{T}$ with parameters $p=q \in (0, \infty]$ and smoothness $s \in \mathbb{R}$.
For $p>0$ fixed and $f …
- 2,306
0
votes
Accepted
Critical Smoothness on Besov Spaces $B^s_{p}$: how does it evolved with $p$?
The answer to my question is actually no: there exists generalized functions such that $s_p(f)$ is not of the proposed form. Stéphane Jaffard discusses the possible forms of the functions $s_p(f)$, de …
- 2,306
5
votes
0
answers
267
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Riesz Representation Theorem for $L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$?
The spaces $L^2(\mathbb{R})$ (square-integrable functions) and $L^2(\mathbb{T})$ (1-periodic square-integrable functions, considered over the real line $\mathbb{R}$) are two subspaces of the space of …
- 2,306
5
votes
2
answers
340
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For which tempered distributions is the fractional derivative well-defined?
Let $\gamma \geq 0$ and consider the fractional derivative operator defined in Fourier domain by
$$\mathcal{F} \{\mathrm{D}^{\gamma} \varphi \} (\omega) = (\mathrm{i} \omega)^{\gamma} \mathcal{F}\{\v …
- 16.2k
4
votes
2
answers
797
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Completion of $\mathcal{S}(\mathbb{R})$ for a given norm
Assume that $\lVert \cdot \rVert$ is a norm on the space of rapidly decaying functions $\mathcal{S}(\mathbb{R})$. Under which conditions on the norm can we say that the completion $\mathcal{X}$ for th …
- 3,085
2
votes
0
answers
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Generalized Besov spaces with different integrability and smoothness in space and time?
Consider the family of Besov spaces $B_{p,q}^{s}(\mathbb{R})$ with $0<p,q \leq \infty$ and $s \in \mathbb{R}$.
Is there a natural way to define spaces of generalized functions $f(t,x) \in \mathcal{S} …
- 2,306
5
votes
3
answers
356
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No kernel of the form $\lvert x - y\rvert^{-1}$ on tempered distributions?
Does there exist a continuous bilinear form $\mathcal{B}$ on $\mathcal{S}(\mathbb{R})\times \mathcal{S}(\mathbb{R})$ such that
\begin{equation}
\mathcal{B}(\varphi_1, \varphi_2) =\int_{\mathbb{R}\time …
- 3,085
5
votes
0
answers
178
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Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?
Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mathscr{P}$ be a probability mea …
- 2,306
4
votes
0
answers
639
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Define the space of distributions with algebraic decay?
A tempered distribution $u\in \mathcal{S}'(\mathbb{R})$ is said to be rapidly decreasing if for every $f \in \mathcal{S}(\mathbb{R})$, $u*f \in \mathcal{S}(\mathbb{R})$.
One rough way to motivate t …
- 2,306
9
votes
1
answer
886
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Is the space of rapidly decreasing (non-smooth) functions nuclear?
We denote by $\mathcal{S}(\mathbb{R})$ the space of smooth and rapidly decreasing functions. We define on $\mathcal{S}(\mathbb{R})$ the family of semi-norms
$$\lVert \varphi \lVert_{n,m} = \lVert (1+ …
- 2,306