Finite symmetric convolution operators and singular symmetric differential operators

AbstractIn this note, we prove a generalization of a theorem of Morrison (Notices Amer. Math. Soc. 1962, 119) which states that the only real finite symmetric convolution integral operator K in L2[−1, 1] with kernel ϱ analytic about the origin, which admits a second order, singular symmetric differential operator in its commutator is (aside from a trivial scaling) given by Kφ(x) = ∝−1+1 ϱ(x − y) φ(y) dy, φ ϵ L2[−1, 1], with ϱ(x) = Ω2sinh Ω1xΩ1sinh Ω2x, −2 ⩽ x ⩽ 2, where Ω1and Ω2 are either real or pure imaginary. If either Ω1, or Ω2 = 0, one takes the appropriate limit. The associated commuting differential operator is L = ddx a(x) ddx + b(x), with a(x) = 1 − sinh2 Ω2xsinh2Ω2 and b(x) = (Ω12 − Ω22) a(x) + Ω3. Here Ω3 is an arbitrary real constant. Our result states that this remains true, under the relaxed assumptions that ϱ is C2, with 0 as a non-degenerate critical point, and a(x) and b(x) are C1 and C0, respectively. Thus under a mild smoothness and non-degeneracy hypothesis, the only examples are those in which K is a time-and-band limiting operator and L is the operator originally found by Landau, Pollak, and Slepian