Let $G$ be a topological group. Let $f_{n}$ and $g_{n}$ be two sequences in $L^{p}(G)$ that are convergent to $f$ and $g$, respectively. Let $f * g \in L^{p}$. Is $f_{n} * g_{n}$ convergent to $f * g$? why? ($f_{n}*g_{n}\in L^{p}$ for all $n\in \mathbb N$).
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What does \ast mean ? – Mahdi Khosravi Aug 05 '13 at 10:59
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it's LateX for * – Aug 05 '13 at 11:02
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It's the convolution – Aug 05 '13 at 11:29
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Do you assume that $f_n * g_n$ is also in $L^p$? – abatkai Aug 05 '13 at 13:00
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Yes,$f_{n}*g_{n}$ is in $L^{p}$ for each natural n – Aug 05 '13 at 13:59
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In general, convolution does not map $L^p\times L^p$ to $L^p$. It will however map $L^p\times L^p$ to $L^{\frac{p}{2-p}}$. – Cameron Williams Aug 06 '13 at 01:24
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I think Young's inequality works here (but it has been a while...) – Potato Aug 06 '13 at 02:11