Abstract: A fundamental question in the theory of generalized convolutions is the following: given an elliptic operator L on a domain E of Euclidean space, can we construct a convolution-like operator which commutes with L, in analogy with the corresponding property between the ordinary convolution and the Laplacian? It turns out that a positive answer to this question depends on certain geometrical properties of the eigenfunctions of the elliptic operator. In this talk we introduce some basic notions and facts from spectral theory and differential operators. Then we show that if the domain E is smooth and bounded, then the Laplacian eigenfunctions on E do not have a common critical point, and we discuss the implications of this result for the motivating question above.