Stenger’s conjecture on the spectrum of the Sinc convolution operator J

Establish whether the spectrum σ(J) of the collocation operator J arising from the SE–Sinc indefinite integration (the Sinc convolution approximation to the integral operator ℐ on [a,b]) is entirely contained in the open right half–plane Ω+ = {z ∈ ℂ : Re z > 0}, i.e., prove σ(J) ⊂ Ω+, so that the matrix function F(J) is well-defined for functions F analytic on Ω+.

Background

The Sinc convolution approximates an indefinite convolution p(x) by expressing it as (F(ℐ)g)(x) and then replacing the integral operator ℐ with a finite-dimensional collocation operator J built from the SE–Sinc indefinite integration. Defining F via the Laplace-transform-based map F(s) = \hat{f}(1/s), the well-definedness of F(J) requires that the spectrum of J lie in a region where F is analytic.

Stenger conjectured that σ(J) lies in the open right half-plane Ω+, which would ensure F(J) is well-defined under the usual assumption that F is analytic on Ω+. Despite numerical evidence for finite sizes, a general proof has not been established; the paper notes this conjecture has remained open and references partial progress that did not fully resolve it. The present work gives a different resolution by avoiding the need for σ(J) ⊂ Ω+, but the original spectral inclusion conjecture itself is not fully proved.