Motivated by this question, I am interested in the following:
Let $\,(x_n^1)_{n \geq 1}$, $(x_n^2)_{n \geq 1}$, $(x_n^3)_{n \geq 1}$, $\,\ldots\, $ be an orthonormal sequence of square-integrable real sequences $(x_n^i)_{n \geq 1} \in l^2(\mathbb{R})$.
(By "orthonormal", I mean orthonormal with respect to the standard dot product $\langle (x_n),(y_n)\rangle = \sum_n x_ny_n$.)
It is not hard to show that for each $n \geq 1$, $x_n^i \to 0$ as $i \to \infty$.
Are there any results guaranteeing some order of rate of convergence of $$ \sum_{n=1}^\infty \frac{|x_n^i|}{2^n} $$ to $0$ as $i \to \infty$?
