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What is the decay rate of DPSS sequences in frequency?

Consider an interval $T\subset\mathbb{Z}$ of length N in time. Consider another interval $[-W,W]$ in frequency with $W<1/2$. Let $\phi_0$ denote the top eigen-sequence of $P_I P_{[-W,W]}$. Here $P_I$ is the projection onto $I$ and $P_{[-W,W]}$ projects the discrete-time Fourier transform (DTFT) of a signal onto $[-W,W]$ and then takes the inverse DTFT. (In other words, $P_{[-W,W]}$ limits the spectrum of a signal to $[-W,W]$.) From the seminal work of Slepian, we know that $\phi_0$ is supported on $T$ and maximally concentrated on $[-W,W]$ (in $\ell_2$ sense).

Simulations in MATLAB suggest that the DTFT of $\phi_0$ decays exponentially fast outside $[-W,W]$ and has a slower decay inside $[-W,W]$. Is there anyway to prove something along these lines?

Ideally, I wish to know the decay rate of the top eigen-sequence of $P_I P_{[-W,W]}$ where $I$ is this time a union of intervals in $\mathbb{Z}$.

This would be my first question on mathoverflow and I'd really appreciate any input. Thanks in advance.

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