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      Questions tagged [schwartz-distributions]

      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.

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      Characterizing geometrically Schwartz Kernels of pseudodifferential operators on a compact manifold

      Let $M$ be a compact smooth manifold without boundary. Define $\mathcal{P} \subset \mathcal{D}^{'}(M \times M)$ to be the smallest linear subspace of the space of distributions on the product which is:...
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      Defining the value of a distribution at a point

      Let $\omega\in D'(\mathbb R^n)$ be a distribution and $p\in \mathbb R^n$. If there is an open set $U\subset \mathbb R^n$ containing $p$ such that $\omega|_U$ is given by a continuous function $f\in C(...
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      82 views

      Interchanging Integration Order involving Fourier Transform

      $$f(\omega,u):=\frac1{\omega+iu}$$ where $i$ is the imaginary unit number. We see that the integral of a Fourier transform $$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...
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      86 views

      Fourier inversion formula for compactly supported distributions

      I know that the Fourier transform of a compactly support distribution $u\in \mathscr{E}'(\mathbb{R}^{n})$ is smooth and also satisfies $$ |\hat{u}(\xi)|\leqslant C_{N}(1+|\xi|)^N,\label{1}\tag{1} $$ ...
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      Fourier transform of a Lorentz invariant generalized function

      Consider on $\mathbb{R}^{n+1}$ the indefinite quadratic form defining the Minkowski metric $$B(p)=(p^0)^2-(p^1)^2-\dots-(p^n)^2.$$ Let $\mu$ be a generalized function on $\mathbb{R}^{n+1}$ which is ...
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      Fourier Restriction: extension operator identity

      Consider the extension operator: $$ Eg(x)=\int_S g(\xi)e^{2\pi i x\cdot \xi}d\sigma(\xi). $$ For simplicity we consider the 2D-case, where $S$ is the paraboloid $\xi\mapsto \xi^2$, $\xi\in [-1,1]$. (...
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      Extension of Valdivia-Vogt isomorphism from $\mathscr{D}(K)$ to $\mathscr{E}'(K)$

      Let $M$ be a $d$-dimensional (say, Hausdorff, paracompact, connected and oriented) smooth manifold, and $K\subset M$ compact with $\mathring{K}\neq\varnothing$. M. Valdivia has shown (based on ...
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      45 views

      Entire analytic functions with entire analytic Fourier transform, and corresponding distributions

      I'm interested in the Fourier transform on a space of distributions that includes more than the usual tempered distributions, and in particular allows for $\delta$-distributions supported at complex ...
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      2answers
      79 views

      Delta-distribution composed with a function from the Fourier representation

      A well known representation of Dirac's delta-distribution is via the Fourier transform of distributions: \begin{equation} \delta[f]:=f(0)=\int_{\mathbb{R}}\int_{\mathbb{R}} e^{\mathrm{i}xk}f(k)\mathrm{...
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      Convolution with Schwartz class function

      Let $f, g\in \mathcal{S}(\mathbb R)$ (Schwartz class function), $\delta_0$ (dirac delta distribution). Consider distribution as follows: $$H(x, y)= f(x)g(x)\delta_0(y)-f(y)g(y)\delta_0(x), \ (x, y\...
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      is this explicit linear operator hypo-elliptic

      Consider an operator of the form $$L(\phi):=\Delta \phi + \gamma \phi_{rr}$$ here the $r$ denotes derivative with respect to the radial variable (we are in $ R^N$ say where $N \ge 3$). I am ...
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      1answer
      81 views

      Is convolution jointly continuous on $\mathcal{E}'$?

      Let $\mathcal{E}'(\mathbb{R})$ be equipped with its usual strong topology (being the dual space of $\mathcal{E}(\mathbb{R})$). Is convolution jointly continuous on $\mathcal{E}'(\mathbb{R})$?
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      283 views

      Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$

      I asked this question on Math StackExchange, but it did not receive an answer, despite my offering a bounty to attract attention. I am unsure whether it is appropriate for this venue, but I thought ...
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      1answer
      51 views

      Covergent net in $\mathcal{E}'(\mathbb{R})$ implies bounded?

      Let $\mathcal{E}(\mathbb{R})$ be the space of all $C^\infty$ functions on $\mathbb{R}$ with its usual topology, and $\mathcal{E}'(\mathbb{R})$ be the dual space with the weak* topology. Let $(T_i)_{...
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      46 views

      Dense versus sequentially dense in $\mathcal{E}’$

      Endow the dual space $\mathcal{E}’$ of smooth functions $C^\infty$ (with its metrizable topology described by uniform convergence on compacts for convergent sequences) with the weak* topology. Let $D$ ...

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