A sampling theorem for transforms with discontinuous kernels
Mahmoud Hamed Annaby, Gerhard Freiling · Applicable Analysis · 2004
We present a sampling representation for the transform where the kernel is a solution of an nth order differential equation ℓ (y) = λy. Here the differential expression ℓ is defined to be ℓ = ℓ1 on [−1,0) and ℓ = ℓ2 on (0,1], ℓ1 and ℓ2 are in general two different nth order differential expressions. This includes the case when ℓ1 and ℓ2 are identical with discontinuous coefficients at x = 0. So, in addition to the boundary conditions, the eigenvalue problems in question contain n compatibility conditions. The problem is not necessarily self adjoint, but conditions are imposed on it in which the eigenfunctions are a Riesz basis.