A lower bound for an exponential series

Let $\gamma >0$ and $r=(r_n)_{n \in \mathbb{Z} } \in \ell^\infty(\mathbb{Z})$ (a bounded and real-valued sequence), where $r_n \geq 0 \ \forall \ n \in \mathbb{Z}$. Consider the function $f(x)=\sum\limits_{n \in \mathbb{Z}} r_n \exp \left (-2 \gamma \left (x- \frac{n}{2} \right)^2\right)$ for $x \in \mathbb{R}$.

 

Prove that there exists a constant $C>0$ such that $\sup\limits_{x \in \mathbb{R} }|f(x)| \geq C \cdot ||r||_\infty$. 

  • The statement doesn't make much sense. You are taking the supremum is the left had side over all x \in R and they say for all x \in R. Did you mean having just f(x) without supremum in the left hand side of what you wish to prove?

  • yes my bad, I corrected it. thanks

  • I still think "sup" should be replaced by "inf" for the problem to make sense.

  • Proving the statement with "sup" is trivially easy.

  • For inf it is equally trivial.

  • would it be equally trivial if I didn't require each r_n to be non-negative?

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