I am looking for a suitable way to define bandlimited functions on a noncompact Riemannian manifold. If it simplifies the problem, assume that the manifold is complete and allowed to have a doubling property also.
In the Euclidean sense, when one has a bandlimited function $f$ with $\hat{f}$ supported on $B_R(0)$, one can write $f(y) = (f \ast \phi)(y) = \int_{\mathbb{R}^n} f(x) \phi(y-x) \, dx$ via a cut-off function $\hat{\phi}$ supported on $B_R(0)$. In other words, $f$ can be expressed as an integral transform.
Is there a definition suitable definition for bandlimited on a noncompact manifold that allows me to write my function $f$ in the form $\int f(x) K(x,y) \, dx$ for some kernel $K$?