Let $\{ g_n \} $ be a frame in a separable Hilbert space $H$. Then the frame operator $S:H\to H$ defined as \begin{equation} S f := \sum_{n=1}^\infty (f,g_n)g_n \end{equation} is a Hilbert space isomorphism i.e., continuous and bijective. Then one can define the canonica dual frame of $g_n$ as the frame $S^{-1}g_n=:f_n$. It is quite straightforward to prove that this is in fact a frame. Where I am having trouble is the following claim from the artile "Revisiting Landauʼs density theorems for Paley–Wiener spaces" in page 5 the authors claim that furthermore $|\langle f_n,g_n \rangle | \leq 1 $. The reference that they give does not give a proof of this claim. A proof is given in Lemma 6.14 of "What is variable bandwidth", but to me this (two lines) proof is not transparent.

I still think that the claim holds, but I was not able to prove it.