Eigenfunctions of Covariance Operators

In the context of discussing the eigenfunctions of a linear integral operator,
especially in the field of functional analysis and its applications to
differential equations or statistical processes (like covariance operators in
stochastic processes), a commonly used symbol for eigenfunctions is $\psi$
(psi). This choice is largely a matter of convention and clarity, allowing for
easy distinction from other functions or variables within the same
mathematical framework.

When considering a linear integral operator, often denoted by $K$, acting on a
function $f$ over some domain $D$, the integral equation representing the
action of $K$ on $f$ can be written as:
\begin{equation}
  (Kf) (x) = \int_D R (x, y) f (y)  \hspace{0.17em} dy
\end{equation}
where $R (x, y)$ is a kernel function characterizing the operator. The
eigenfunctions $\psi_n$ of this operator satisfy the equation:
\begin{equation}
  (K \psi_n) (x) = \lambda_n \psi_n (x)
\end{equation}
where $\lambda_n$ represents the eigenvalue associated with the eigenfunction
$\psi_n$. This formulation is crucial in many areas, including solving
integral equations, quantum mechanics, and in the spectral theory of
operators, where eigenfunctions play a key role in expanding functions in
terms of the operator's eigenbasis.