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Let $q = \{q_n\}_{n\in\mathbb{Z}}$ be a real sequence and write $\mathcal{N} := \sum\limits_{i\in\mathbb{Z}}|q_i|^2$ \begin{align} \sum_{n\in\mathbb{Z}}\sum\limits_{\substack{m\in\mathbb{Z} \\ m\neq n}}\frac{|q_n - q_m|^2}{|n-m|^{1+\alpha}} &\leq \sum_{n\in\mathbb{Z}}\sum\limits_{\substack{m\in\mathbb{Z} \\ m\neq n}}\frac{|q_n|^2 + |q_m|^2}{|n-m|^{1+\alpha}}\\ &= \sum_{n\in\mathbb{Z}}|q_n|^2 \left ( \sum\limits_{\substack{m\in\mathbb{Z} \\ m\neq n}}\frac{1+|q_m|^2}{|n-m|^{1+\alpha}} \right )\\ &= \zeta(1+\alpha)\sum_{n\in\mathbb{Z}}|q_n|^2 + \zeta(1+\alpha)\sum_{n\in\mathbb{Z}}|q_n|^{2}\sum\limits_{\substack{m\in \mathbb{Z} \\ m \neq n}}|q_m|^2\\ &?= \zeta(1+\alpha) \left ( \mathcal{N}\ + \mathcal{N}^2 \right ) \end{align}
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Mon, 15 May 2023 18:29 GMT