Tagged Questions

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A short question concerning the distributional solution of $xf=0$

I was reading my notes on the following result: All the $\mathcal{D}'(\mathbb{R})$ solutions to $xf =0$ are of the form $c\delta$ where $c$ is constant and $\delta$ is the dirac delta distribution ...
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$xT' = 1$ in $D'(\mathbb{R})$

I need help solving the following problem: I want to show that all solutions of $$xT' = 1\ , T \in D'(\mathbb{R})$$ take the following form: $c_{1} + c_{2}1_{[0, \infty)} + ln|.|$ What I tried so far ...
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Translation is continuous

Let $\mathcal D$ be the space of 'test-functions'. Those are infinitely differentiable functions with compact support. Define the following convergence on $\mathcal D$. $(\phi_j) \to \phi$ in ...
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Fourier Transform of Dirac Comb on $\mathbb{Z}$ and $\mathbb{Z}^{d}$.

Let $f(x)=\sum_{n\in\mathbb{Z}}\delta(x-n).$ (a) Show $f$ is a tempered distribution. (b) Compute $\hat{f}$ using the convention $\int_{\mathbb{R}}f(x)e^{-ix\xi}\;dx$ convention for $\mathcal{F}$. ...
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Constructing a Distributional Solution to the Inhomogeneous C.R. Equations

The question is to find a fundamental solution to the system of equations in $\mathbb{R}^{2}$ \begin{array}{l} u_{x}-v_{y}=f\\ u_{y}+v_{x}=g\end{array} and to express the answer as a $2\times2$ ...
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For $1 \leq p < \infty, n\geq 1$ my guess of the answer was $W^{1,p}(\mathbb{R}^n)$ but I can't prove the inclusion $\overline{C^\infty_c(\mathbb{R}^n\setminus\{0\})} \subseteq ... 0answers 28 views $u \in H^n$then also$|u|^2 \in H^n$for complex valued function in Sobolev space Suppose that a complex valued function$u$is in the Sobolev space$H^n(R^n) = W^{n,2}$. Is it necessarily true that$|u|^2 \in H^n$? The result is true for real - valued functions since we know ... 1answer 118 views Convergence of a integral - heat Kernel and dirac delta function Consider$\varphi \in S(R^n)$(space of rapidly decreasing functions). Consider the heat kernel $$K_t(x) = \displaystyle\frac{1}{{(4\pi t)}^{n/2}} \displaystyle e^{- \displaystyle\frac{|x|^2}{4t}}, ... 1answer 25 views For every A\in \mathcal{L}(C^\infty(\mathbb T^n)) exists k_A\in C^\infty(\mathbb T^n\times \mathbb T^n) such that.. does anyone know whether it is true that for every A\in\mathcal{L}(C^\infty(\mathbb T^n)) there exists k_A\in C^\infty(\mathbb T^n\times \mathbb T^n) such that,$$ Af(x)=\int_{\mathbb T^n} k_A(x, ... 1answer 132 views $ \frac{\partial^2 T}{\partial x\partial y} = 0 $, then$ T = ? $Can we characterize all distributions$T \in \mathcal{D}'(\mathbb{R}^2) $with the following property of distribution derivatives ? $$\frac{\partial^2 T}{\partial x\partial y} = 0$$ For functions it ... 0answers 37 views Convolution Kernel.. does anyone what is the definition of the convolution kernel of an operator, $$A: C^\infty(\mathbb T^n)\rightarrow \mathcal{D}^{'}(\mathbb T^n),$$ where$\mathcal{D}^{'}(\mathbb T^n)$is the space ... 1answer 56 views Schwartz kernel theorem in the case the distributions are induced by smooth functions.. how can I show that if$A:C^\infty(\mathbb T^n)\rightarrow C^\infty(\mathbb T^n)$is a continuous linear operators then there is a unique linear and continuous operator$K_A: C^\infty(\mathbb ...
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I'm trying to understand a computation in my physics script. To describe the Deltadistribution $\delta(x)$ correctly we would need the formalism of distributions, but one can also much less ...
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Uniformly convergence of delta sequences

Let $(f_n)$ be a sequence of continuous functions $f_n: \mathbb R \rightarrow \mathbb R$ such that $$\lim_{n\rightarrow \infty}\int_{\mathbb R} f_n(x) \phi(x) dx=\phi(0)$$ for each continuous ...
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