Analysis of band-limited functions on quantum graphs

A notion of band-limited functions is introduced in terms of a Hamiltonian on a quantum graph Γ. It is shown that a band-limited function is uniquely determined and can be reconstructed in a stable way from a countable set of “measurements” {Φi(f)}, i∈N, where {Φi} is a sequence of compactly supported measures whose supports are “small” and “densely” distributed over the graph. In particular, {Φi}, i∈N, can be a sequence of Dirac measures δxi, xi∈Γ. A reconstruction method in terms of frames is given which is a generalization of the classical result of Duffin–Schaeffer about exponential frames on intervals. The second reconstruction algorithm is based on an appropriate generalization of average variational splines to the case of quantum graphs. To obtain all these results we establish some analogs of Poincaré and Plancherel–Polya inequalities on quantum graphs.