Prolate spheroidal wave functions

Remarks

Example: for $c = 0$ the equation is Legendre's equation $((1 - x^2)\psi')' = -\chi \psi$. Its bounded solutions are the Legendre polynomials $P_m$ with $\chi_m = m(m + 1)$, so $\psi_m = \sqrt{(2m + 1)/2}\, P_m$; for instance $\psi_0 = 1/\sqrt{2}$ satisfies the equation with $\chi_0 = 0$.

Non-example: for $c = 0$ and $\chi = 0$, the function $\psi(x) = \frac12 \log \frac{1 + x}{1 - x}$ also solves the equation, since $(1 - x^2)\psi'(x) = 1$. It is not bounded near $\pm 1$, so it is not a prolate spheroidal wave function.

For $c > 0$, the functions $\psi_m$ are also the eigenfunctions of the integral operator $$Q_c \psi(x) = \int_{-1}^1 \frac{\sin c(x - y)}{\pi (x - y)}\, \psi(y)\, dy$$ on $L^2(-1, 1)$, which is self-adjoint, positive and compact. Its eigenvalues satisfy $1 > \lambda_0 > \lambda_1 > \cdots > 0$. The kernel of $Q_c$ is that of the projection onto functions whose Fourier transform, in the convention $\int f(t) e^{-i\omega t}\, dt$, vanishes outside $[-c, c]$. Extending $\psi_m$ to $\R$ by the same integral, divided by $\lambda_m$, gives a band-limited function, and $\lambda_m$ is the fraction of its energy that lies in $[-1, 1]$.