# Tagged Questions

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### When sequentially continuous linear functional is continuous?

Let $C^\infty(X)$ denote the space of infinitely smooth functions on a compact manifold $X$ (at the beginning one may assume that $X$ is a circle, though I need a more general case). Let ...
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### Linear operators on distributions with different topologies

Denote by $\mathscr{D}^\prime$ and $\mathscr{D}^\prime_b$ the space of distributions on $\mathbb{R}^n$ equipped with the weak and the strong topology, respectively. Because the topology of ...
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### Convergence of Schwartz Kernels

I read this question, and I would like to ask the opposite: Assume that I have a sequence of smoothing operators $(P_n)$ with (hence smooth) kernels $(p_n)$ converging strongly to some smoothing ...
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### smoothing a current

Let $M$ be a smooth oriented manifold of dimension $n$ and $T$ a current of dimension $k$ on $M$. Let $\phi:P\times M \to M$ be a proper smooth family of diffeomorphisms of $M$ (i.e. $P$ is a smooth ...
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### Zero currents localized along a submanifold

Let $\mathcal{D}(\mathbb{R})$ be the continuous dual of $C^\infty_c(\mathbb{R})$, the space of compactly-supported smooth functions. There is a nice characterization of distributions ...
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### The derivative of a non-tempered distribution can be tempered?

Suppose we have a non- tempered distribution $u\in \mathcal D'(\mathbb R^d)\backslash \mathcal S'(\mathbb R^d)$. Is it possible to have $\partial_{x_1}...\partial_{x_d}u \in \mathcal S'(\mathbb R^d)$ ...
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### Estimate the analytical wavefront set $WF_A(u)$ given $WF_A(A_K u)$

Let $X$ and $Y$ be two real-analytic manifolds and $Z \subset X \times Y$ be a real-analytic embedded closed submanifold. Suppose now that $K \in \mathscr D'(X \times Y)$ is a distribution conormal to ...
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### Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$

Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$. When $f$ is a function, it looks like the only solution is $f(x)=0$. But what if we allow distributions, such as the Dirac delta?
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### Fourier transform of tempered distribution

I'm wondering whether anyone knows a reference or proof for finding the Fourier transform of $f(t):=(t+1)^{1/2}t_+^{1/2}$? (Here $t_+=\max (t,0)$.)
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### Nonlinear PDE and Green functions

This is somewhat of a curiosity that can hide somewhat deeper. For a Green function of a nonlinear PDE I mean something like $$\partial^2\phi+V(\phi)=\delta^D(x).$$ I do not know if a real ...
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### Fourier transform of a bounded function

This should really be well-known, but I was not able to find a definite answer to this question: Is the Fourier transform of a bounded function always a borel measure (i.e. an order 0 distribution)? ...
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### Regularized fractional derivative of distributions.

A fractional derivative of distributions is usually introduced using definition of fractional integral as a convolution of two distributions. However there is another approach based on fractional ...
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### Inverse schwartz-distribution for convolution operation

I note here $\mathcal{D}'$ the space of all distributions and $\mathcal{S}'$ the space of tempered distributions, I am considering the following question: Let $u \in \mathcal{D}'$ or $\mathcal{S}'$, ...
Is there some regularizing version of Schwartz kernel theorem for topological spaces, i.e., in the form of Every continuous linear map $A\colon C\prime(X_2) \to C(X_1)$ is given by a kernel $k \in ... 1answer 130 views ### Distributional limits concerning the regularity of Maxwells equations This question is related to my previous question about the regularity of the Maxwell equations. Assume we are working on a space where there are only electric point charges,$(q_i)$, and a blob of ... 4answers 349 views ### Reference for integral of functions taking values in a topological vector space. (Note that I am interested in the Gelfand-Pettis integral specifically, as opposed to, for example, the Bochner integral.) I have tried Googling things like "integral topological vector space", ... 2answers 304 views ### Eigenfunction of local fractional derivative Let$E_{\alpha}(x^{\alpha})$be a Mittag-Leffler function,$\alpha \in (0,1)$. It is an eigenfunction for nonlocal fractional derivative, defined as a convolution with$$\Phi_{\lambda}(x) = ... 2answers 401 views ### Fourier transform of$e^{it|\xi|^{\alpha}}$Consider the fourier transform of$e^{it|\xi|^{2\alpha}}$($\alpha>0$)in$\mathbb{R}^n$,let$K_{\alpha}=\mathcal{F}(e^{it|\xi|^{\alpha}})$,so$K$is a tempered distribution.Now I want to know if ... 1answer 487 views ### What functions can be obtained as a convolution of a Schwartz function and a tempered distribution? Let$\mathcal S (\mathbb R)$denote the space of Schwartz functions on$\mathbb R$and$\mathcal S^* (\mathbb R)$denote the dual space of Schwartz (a.k.a tempered) distributions. We consider ... 1answer 408 views ### Calculating a distributional derivative Suppose that we have a sequence of functions$u_j$that are in$L^{\infty}(0,1)$. Then the sequence of maps$N_j(s) := \|u_j(s)\|^2$are also in$L^{\infty}(0,1)$. Hence they give rise to ... 2answers 378 views ### Opinions on the Multiplication of Measures A few questions, hopefully to spark some discussion. How can one define a product of measures? We could use Colombeau products by embedding the measures into the distributions? I'm not sure why ... 2answers 399 views ### One-sided Cauchy principal value What is the notion of a principal value of an integral when the singularity appears at one endpoint? Namely,$ PV \int_a^b f(t) dt = ? $, where the integral is convergent in the upper limit, but ... 2answers 269 views ### Integration under functional sign Let$f(x,y)$be some bounded with its derivatives continuous function on$\Omega \times \overline{\Omega}$, where$\Omega$is a domain in$\mathbb{R}^n$. Let$f(\,\,\cdot\,,\,y) \in ...
When we have a distribuion $u\in \mathcal{D}'(R^n)$,and the restriction to $R^{n}\backslash{0}$ is homogeneous of degree a,we have $u \in \varphi'$ and $\widehat u$ is of degree(-n-a) in ...