Linear ordinary differential equations with boundary conditions on arbitrary point sets

Abstract:

Boundary-value problems for differential operators $\Lambda$ of order $2m$ which are the Euler derivatives of quadratic functionals are considered. The boundary conditions require the solution $F$ to coincide with a given function $f \in {\mathcal {H}_L}(R)$ at the points of an arbitrary closed set $B$, to satisfy at the isolated points of $B$ the knot conditions of $2m$-spline interpolations, and to lie in ${\mathcal {H}_L}(R)$. Existence of solutions (called “$\Lambda$-splines knotted on $B$") is proved by consideration of the associated variational problem. The question of uniqueness is treated by decomposing the problem into an equivalent set of problems on the disjoint intervals of the complement of $B’$, where $B’$ denotes the set of limit points of $B$. It is also shown that $\Lambda$, considered as an operator from ${\mathcal {L}_2}(R)$ to ${\mathcal {L}_2}(R)$), with appropriately restricted domain, has a unique selfadjoint extention ${\Lambda _B}$ if one postulates that the domain of ${\Lambda _B}$ contains only functions of ${\mathcal {H}_L}(R)$ which vanish on $B.I + {\Lambda _B}$ has a bounded inverse which serves to solve the inhomogeneous equation $\Lambda F = G$ with homogeneous boundary conditions. Approximations to the $\Lambda$-splines knotted on $B$ are constructed, consisting of $\Lambda$-splines knotted on finite subsets ${B_n}$ of $B$, with $\cup {B_n}$ dense in $B$. These approximations ${F_n}$ converge to $F$ in the sense of ${\mathcal {H}_L}(R)$.