# Questions tagged [generalized-functions]

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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 ...
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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 & ...
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### When can we integrate a distribution over a smooth domain?

Maybe we can restrict the discussion to one dimension: When can we integrate a distribution over an interval? In Lebesgue theory, we can integrate measurable functions. But distributions are not ...
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### Colombeau generalized functions

I'm currently reading some aspects of Colombeau generalized functions, and in almost all of his examples he discuss aspects of Quantum Field Theory, but then I go to some "standard" texts on QFT and I ...
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### Total variation of Dirac's delta function

This question on math (dot) stackexchange (dot) com has inspired me to write my own (possibly somewhat tendentious?) version of the question. What does the question mean? That was a topic of a lot ...
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### Non-compact analogue of Peter-Weyl

I have the following situation: $G$ is a real unimodular locally compact semisimple Lie group. Then it is known that the regular representation $H:=L^2(G,\mu_H)$ decomposes as \begin{equation} \int^{\...
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### The abstract kernel theorem implies Schwartz kernel theorem. How exactly?

Let me first give a little rapid background prior to formulating the question. Let $\mathcal{D}$ be a Schwartz space of infinitely differentiable functions and $\mathcal{D}'$ is the space of ...
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### Hyperfunctions supported at a point

Is it true that the space of hyperfunctions on $\mathbb{R}^n$ supported at 0 coincides with the space of Schwartz distributions supported at 0? More explicitly, is it true that any hyperfunction ...
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### Applications and main properties of hyperfunctions

I am trying to get familiar with hyperfunctions, and I do have some familiarity with the classical theory of distributions. I am wondering whether hyperfunctions have any advantages over ...
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Let $T \in \mathscr D'(\mathbb R^n)$ be a distribution, such that its Fourier transform $\widehat T$ is a real analytic function on $\mathbb R^n$ but it can't be continued to an entire function on $\... 2answers 991 views ### Wiener measure and Bochner Minlos I am reading probability theory and I have a question. The Bochner-Minlos theorem roughly says that if we have$E \subset H \subset E^*$where$H$is a Hilbert space, then there is a unique measure ... 1answer 366 views ### How to define a generalized differential form through its values on submanifolds Suppose we're in$\mathbb R^n$, and we have a function on line segments ,$\omega(I)$, with values in$\mathbb R$. Give sufficient conditions for$\omega$to be given by a generalized 1-form (that is, ... 1answer 298 views ### Sequential continuity of linear operators Let$u\colon L\to M$be a linear map of locally convex linear topological vector spaces. Assume that$u$is sequentually continuous, i.e. maps convergent sequences to convergent ones. (This notion is ... 3answers 166 views ### sequences of plane measures converging to a singular one: terminology, etc We are dealing with very "easy" sequences of uniform measures converging to singular measures (?), as in the following example: let$a$,$b$, and$c$be vertices of a triangle in$\mathbb{R}^2$, and$...
Let X,Y be smooth manifolds and let $f:X \to C^{-\infty}(Y)$ be a continuous map, where $C^{-\infty}(Y)$ is the space of generalized functions on $Y$ equipped with the weak topology. By Schwartz ...