Given a sequence $\{d_{m}\}_{m\in{\bf{Z}}^{n}}$ of nonnegative numbers with $d_{m}\rightarrow 0$ as $|m|\rightarrow\infty$, I am looking for a sequence $\{a_{j}\}_{j\in{\bf{Z}}}$ such that the product $a_{m_{1}}\cdots a_{m_{n}}\geq d_{(m_{1},...,m_{n})}$ and $a_{j}\rightarrow 0$ as $|j|\rightarrow\infty$, does anyone here know how to construct explicitly such a sequence $\{a_{j}\}_{j\in{\bf{Z}}}$?
Choosing a dominated sequence with certain properties
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real-analysis
sequences-and-series
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0What do you mean by $a_{m_1}\cdots a_{m_n}$? Is that the product? And what does $d_{(m_1,...,m_n)}$ mean ? – 2017-01-14
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0The first one is product of numbers. For the $(m_{1},...,m_{n})$, note that $m=(m_{1},...,m_{n})$ is a $n$-tuple of integers. – 2017-01-14
1 Answers
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I suppose $|(m_1, \dots, m_n)|:= \max_{i=0}^n |m_i|$.
Then we can take an increasing sequence $(M_k)_{k\in\mathbb{N}}\subset\mathbb{N}$ s.t. \begin{eqnarray} |m|\geq M_k \Rightarrow d_m < \frac{1}{k^n}. \end{eqnarray}
Let $X:=\max\{1, \max_{m\in\mathbb{Z}^n}d_m\}^{1/n}$ and put
\begin{eqnarray}
a_j :=
\begin{cases}
X & (|j|