Discontinuous boundary-value problems: expansion and sampling theorems (Q2567961)

The authors consider a linear boundary value problem in \([-1,1]\) of the form \(l(y)+\lambda y=0\), where \(l\equiv l_1\) in \([-1,0]\) and \(l\equiv l_2\) in \([0,1]\) are two \(n\)th-order linear differential operators, together with \(n\) boundary conditions given by linear forms on \(y^{(j)}(\pm1)\) and \(n\) compatibility conditions at 0 given by linear forms on \(y^{(j)}(0^\pm)\), \(0\leq j\leq n-1\). Only strongly regular problems are considered, in the sense that the asymptotics of a characteristic determinant \(\Delta(\lambda)\) as a function of \(\lambda\) in the plane satisfies some nondegeneracy assumption. Versions of classical theorems are then obtained, e.g., the expansion of Green's function as a uniformly convergent series in terms of products of eigenfunctions and the expansion of any function satisfying the boundary and compatibility conditions in terms of eigenfunctions. The main sampling theorem gives a certain integral transform \(F(\lambda)\) of an arbitrary \(L^2(-1,1)\)- function \(f(x)\), namely \[ F(\lambda)=\int_{-1}^1\Phi(x,\lambda)\bar f(x)\,dx;\;\;\;\Phi(x,\lambda):=\Delta(\lambda)G(x,\xi_0,\lambda) \] (\(G\) is Green's function of \(l-\lambda\)), in terms of the locally uniformly convergent series \[ F(\lambda)=\sum_{k=1}^\infty F(\lambda_k)\frac{\Delta(\lambda)}{(\lambda-\lambda_k)\Delta'(\lambda_k)} \] (\(\lambda_k\) are the eigenvalues). The paper is complemented with some examples involving first- and second-order operators.