I would like to know if the Prolate Spheroidal Wavefunctions (PSWFs, defined below) are in $L^1(\mathbb{R})$. I know that they are square integrable, but cannot decide about absolute integrability.


The Prolate Spheroidal Wave Functions are eigenfunctions of the following integral equation:
$$\int\_{-T}^T\varphi\_n(x) \text{sinc}(t-x) dx = \lambda\_n \varphi\_n(t)$$


where $\text{sinc}(t) = \sin(\pi t)/ \pi t$. Alternatively (as discovered by Slepian et al.) they are also the eigenfunctions of the following differential operator:
$$(1-t^2)\frac{d^2\varphi\_n}{dt^2}-2t\frac{d\varphi\_n}{dt} -(2 \pi T \Omega)^2t^2 \varphi\_n = \mu\_n \varphi\_n$$


The Prolates are bandlimited to $[-\Omega/2, \Omega/2]$ and maximally time-concentrated on the interval $[-T, T]$ (see the series of papers by David Slepian, Landau, and Pollack). As such they are entire functions in the complex variable $t$.



Are the Prolate Spheroidal Wave Functions absolutely integrable?
They are not in $L^1$. The principal term of the asymptotics is
$$\frac{e^{\pm iTx}}{x}.$$
This asymptotics is written for example here:


Richard-Jung, F.; Ramis, J.-P.; Thomann, J.; Fauvet, F.
New characterizations for the eigenvalues of the prolate spheroidal wave equation. (English summary) 
Stud. Appl. Math. 138 (2017), no. 1, 3–42. 


