Prolate spheroidal wave function

The Prolate Spheroidal Wave Functions are a set of functions derived by timelimiting and lowpassing, and a second timelimit operation. Let $Q_{T}$ denote the time truncation operator, such that $x=Q_{T}x$ iff x is timelimited within $[-T/2;T/2]$. Similary, let $P_{W}$ denote an ideal low-pass filtering operator, such that $x=P_{W}x$ iff x is bandlimited within $[-W;W]$. The operator $Q_{T}P_{W}Q_{T}$ turns out to be linear, bounded and self-adjoint. For $n=1,2,\ldots$ we denote with $\psi _{n}$ the n-th eigenfunction, defined as
$Q_{T}P_{W}Q_{T}\psi _{n}=\lambda _{n}\psi _{n}$,
where $\{\lambda _{n}\}_{n}$ are the associated eigenvalues. The timelimited functions $\{\psi _{n}\}_{n}$ are the Prolate Spheroidal Wave Functions (PSWFs).

Sources

I. Daubechies, "Ten Lectures on Wavelets"