TITLE: Upper bound of series
QUESTION [0 upvotes]: I have a series: $$S = \sum_{k=1}^{\infty}\frac{|x(\frac{3}{2k}-1)|^{2}}{k^5}$$ where $x \in C[-1,1]$. How can I bound this series? I think this inequality $$S \leq \max_{t\in[-1,1]}|x(t)|$$ are true, but  I don't know how I can proof this.
Thank you in advance!

REPLY [2 votes]: Not knowing more about $x$, about all you can write is
$$S\le\max_{t\in[-1,\frac12]}x^2(t)\,\zeta(5)=\left(\max_{t\in[-1,\frac12]}|x(t)|\right)^2\,\zeta(5)$$
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