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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.

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I assume your inner product is conjugate linear in the second factor, based on your assertion that the operator $S$ is a Hilbert space iso. Then $$ \langle f_n, g_n\rangle = \langle f_n, S f_n \rangle = \langle f_n, \sum_k \langle f_n, g_k\rangle g_k\rangle = \sum_k |\langle f_n, g_k\rangle|^2 $$ is real and non-negative. Subtract $|\langle f_n,g_n\rangle|^2$ from both sides, you find $$ |\langle f_n,g_n\rangle| - |\langle f_n,g_n\rangle|^2\geq 0$$ which can only hold if $|\langle f_n,g_n\rangle| \leq 1$.

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