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If $u_n \in C_c^\infty (\mathbb R)$ with $u_n=u(x+n)$ , $n\in \mathbb N$ , $u$ is not identically zero.

How do i prove that $||u_{n+k}-u_n||_{L^q}^q =2||u||_{L^q}^q$.

What my doubt is that even if we take $u_{n+k}$ and $u_n$ to have disjoint support, it doesn't seem that the equality holds . $||u_{n+k}-u_n||_{L^q}^q=\int |u(x+n+k)-u(x+n)|^q$ , i don't see how i can relate the value of $u(x+n+k) $ and $u(x+n)$ looking forward for some hints and help. Thanks

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up vote 3 down vote accepted

If the supports of $v$ and $w$ are disjoint it holds that $$ \int |v + w|^q = \int_{\mathrm{supp}(v)}|v+w|^q + \int_{\mathrm{supp}(w)}|v+w|^q. $$ Since $w$ is zero on $\mathrm{supp}(v)$ it holds that $$ \int_{\mathrm{supp}(v)}|v+w|^q = \int_{\mathrm{supp}(v)}|v|^q $$ and vice versa.

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thank you for the help . – Theorem Jul 24 '12 at 9:14

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