# Tagged Questions

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### $f\in L^{p}(\mathbb R)\cap C_{0}(\mathbb R); (1<p<\infty), g\in C^{\infty}_{c}(\mathbb R) \implies f\ast g \in C^{k}(\mathbb R)$?

We put, $C_{0}(\mathbb R)=$ The space of continuous functions on $\mathbb R$ vanishing at $\infty$; $C^{k}(\mathbb R)=$ The space of all functions $\mathbb R$ whose derivative of order $\leq k$ exist ...
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### Estimate of Projection Operator on two-torus

Let $\Lambda$ be a lattice, $\mathbb{T}=\mathbb{R}^2/\Lambda$ be a flat torus and $\Delta$ be the Laplace-Beltrami operator. There is any reference where the norm of the projection operator ...
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### $\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

It is well-known that, for $f,g \in L^{1}(\mathbb R).$ Then, by Fubini's theorem, one can derive, $\widehat{f\ast g} = \hat{f} \cdot \hat{g},$ (that is, Fourier transform takes, convolution to point ...
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### Is Fourier transform density preserving?

I know my question is not well-defined since Fourier domain and codomain are not the same, but one knows that they are actually homomorphic. Now what I mean by density preserving is as follows: ...
### $\int_{\mathbb R^{2}} |\int_{\mathbb R} (f(t-y)- f(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0$ as $y\to 0$?
Fact: It is well-known that translation is continuous in the $L^{1}$ norm, that is, if $f\in L^{1}(\mathbb R)$ then $\lim_{y\to 0} \|f_{y}-f\|_{L^{1}(\mathbb R)}=0;$ (where, $f_{y}(x)= f(x-y)$, ...