Questions tagged [generalized-functions]
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-2
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1
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Non standard representation of rational number as an infinite continued fraction
Let two elementary operations on a real number $y$ be defined by
$$
S_{+}(y):=\frac{1}{,1円+\frac{1}{y},円}=\frac{y}{y+1},
\qquad
S_{-}(y):=\frac{1}{,1円-\frac{1}{y},円}=\frac{y}{y-1},
$$
whenever the ...
4
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0
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183
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Interpreting 1ドル/f$ as a distribution when $f$ is only smooth
My first question is: does there exist a smooth function $f$ such that $f \neq 0$ on $\mathbb{R}^n \setminus \{0\},ドル $f(0) = 0,ドル and 1ドル/f,ドル viewed as a distribution on $\mathbb{R}^n \setminus \{0\},ドル ...
1
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0
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138
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Real interpolation of negative Sobolev spaces $(L^p,W^{-k,p})_{\theta,p}$ for 1ドル\le p\le\infty$
For an open set $U\subset\mathbb R^n,ドル $k\ge1$ and 1ドル\le p\le\infty,ドル the space $W^{-k,p}(U)$ consists of all distributions of the form $\sum_{|\alpha|\le k}\partial^\alpha g_\alpha$ where $g_\alpha\...
3
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1
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Is there a name for this nice property of the usual weak incompressible Navier-Stokes equation?
Navier-Stokes is a non-linear PDE, and there is no standard, general theory of weak solutions for nonlinear PDEs. But the literature on weak solutions to the incompressible Navier-Stokes constantly ...
1
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0
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85
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Embedded branching random walk converge to some random generalized function?
We know that two dimensional discrete GFF(2d-DGFF) on a box $V_N=N[0,1]^2\cap\mathbb{Z}$ with Dirichlet boundary condition will converge in distribution to the 2d continuum GFF with Dirichlet boundary ...
2
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0
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140
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Representation of an operator on a generalized eigenfunction
This is a cross-post from: https://math.stackexchange.com/questions/4651664/representation-of-an-operator-on-a-generalized-eigenfunction
Suppose we have an (essentially) self adjoint operator $L$ ...
2
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1
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240
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Description of $\mathcal{S}^{2}_0(\mathbb{R})$ and certain class of functions inside it
I am currently reading volume 2 of "Generalized Functions" by Gelfand and $\mathcal{S}^{2}_0(\mathbb{R})$ is defined to be the collection of $C^\infty$ functions $f$ on $\mathbb{R}$ such ...
8
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1
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Generalized functions in infinite dimensions
What theories are there for generalized functions (distributions) in infinite dimensions?
In particular, suppose your "infinite dimensional manifold" is $\mathfrak{M}:=C^\infty(S^1)$. Is ...
24
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6
answers
5k
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Anti-delta function?
Did anyone ever consider a "function" or "distribution" $F(x)$ with the following property:
its integral $\int_a^b F(x),円dx=0$ for any finite interval $(a,b)$ but $\int_{-\infty}^\...
1
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0
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150
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product of two generalized functions
Let $f_n$ and $g_n$ two generalized functions such that :
the product $f_n g_n$ is well defined for all $n\in \mathbb{N},ドル which means $WF(f_n)+WF(g_n)$ does not contain an element of the form $(x,0)$...
0
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0
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284
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What's the definition of Euclidean density?
In page 19 of the article "Equivariant cohomology with generalized coefficients" the authors say:
Let $V $ be an n-dimensional real vector space, let $v'$ be a non-zero element in $ \wedge^...
7
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0
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188
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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}(\...
3
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2
answers
240
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What is the distribution of the following limit?
Assume $x \in \mathbb{R}$. We already know that
$$\lim_{\epsilon \to 0+} \frac{1}{x-i\epsilon} - \frac{1}{x+i\epsilon} = 2\pi i \delta_x.$$
Here $\delta_x$ denotes the Dirac distribution. If we ...
8
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1
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767
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English translation of Schwartz's papers on vector-valued distributions
I am interested in systematically studying the theory of vector-valued distributions. The original two papers due to Laurent Schwartz entitled Théorie des distributions à valeurs vectorielles. I & ...
2
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1
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229
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Why does the formula (7) in the article equivariant cohomology with generalized coefficients hold
I'm reading the article of equivariant cohomology with generalized coefficients by Kumar and Vergne and I have this question from that article
Let G be a compact lie group with lie algebra $\mathfrak{...