Bounds on discrete convolution

Question

Let \begin{align} a_n= cos \left( \frac{\pi}{2} n\right)n^{- \frac{1}{2}n +\frac{1}{2} } \end{align}

and let \begin{align} c_{2n}= \sum_{m=0}^{2n} a_m a_{2n-m} \end{align}

Note that the above is an operation of discrete convolution

Question 1 : Can we establish that \begin{align} \lim_{n \to \infty } c_{2n}=0 \end{align}

Question 2: If the about is true can we also establish how fast does $c_{2n}$ go to zero by showing bounds on $|c_{2n}|$?

Thanks you.

Answer 1

Using a Cauchy product :

$$\sum c_n = \left(\sum a_n\right)^2 = \left(\sum \frac{i^n +(-i)^n}{2} n^{- \frac{1}{2}n +\frac{1}{2} }\right)^2$$

which converges, thus $ c_n \longrightarrow 0$ .