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](https://reader.elsevier.com/reader/sd/pii/S1631073X12001392?token=B133329B64EF5D9474F0A08D0637E56463A5D50C211624546E3F3CCE96078C75BF90A4FFE8AD8C78552964F1BA66CE30&originRegion=eu-west-1&originCreation=20230417084448)" 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](https://onlinelibrary.wiley.com/doi/pdf/10.1002/cpa.21694)", 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.



A property of the canonical dual frame in a Hilbert space
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 $|\langle f\_n,g\_n\rangle| \leq 1$.


