A sampling theorem for transforms with discontinuous kernels

Abstract

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.

Keywords:

• Sampling theory
• Expansion in eigenfunctions
• Riesz bases
• Spectral problems with discontinuous coefficients
• AMS Subject Classifications: 41A05
• 34L10
• 34B05

Acknowledgement

The authors wish to thanks Professor W.N. Everitt for discussions and notes concerning the Kramer analytic kernels. The research was supported by AvH-Stiftung under V-8121/AGY/1039259.