1. INTERPOLATORY SOLUTIONS OF LINEAR ODE’S AND EXTENSIONS Wayne M. Lawton Dept. of Mathematics, National University of Singapore 2 Science Drive 2, Singapore 117543 wlawton@math.nus.edu.sg Yongwimon Lenbury Dept. of Mathematics, Mahidol University Rama 6 Road, Bangkok, Thailand 10400

2. its Lie algebra, identified with is a connected Lie group with identity We examine the following initial value problem SCOPE Continuous solution Measure

3. ISSUES  Continuous dependence of solutions  Approximation & interpolation of continuous  Applications and extensions by solutions whereis a dense subspace of the space of-valued measures that vanish on finite sets functions

4. PRELIMINARIES Choose a euclidean structure with norm be the geodesic distance function defined by the induced right-invariant riemannian metric and let

5. PRELIMINARIES space of- valued measures on point masses whose topology is given by seminorms topological group of continuous functionsonthat satisfy equipped with the topology of uniform convergence over compact intervals, under pointwise multiplication - valued without functions having bounded variation locally

6. PRELIMINARIES Lemma 1 is inif and only if then gives the distance along the trajectory in is in A function and

7. PRELIMINARIES subspace of step functions exponential function map control measures to solutions contains dense subset of interpolation set

8. RESULT is dense andTheorem 1 extends to a continuous that is one-to-one and onto. Furthermore, is a subgroup ofand it forms topological groups under both the topology of uniform a homeomorphism. convergence over compact intervals and the finer topology that makes the function

9. DERIVATIONS Lie bracket Adjoint representation for matrix groups We choose such that

10. Lemma 2 If satisfy and then where DERIVATIONS The proof of Theorem 1 is based on the following

11. Proof Apply Gronwall’s inequality to the following

12. RESULT be a dense subspace.Theorem 2 Let Then for every positive integer contains a dense subset of sequences and pair of

13. DERIVATIONS It suffices to approximateby elements in Choose any Lemma 3 Letbe a homeomorphism of a compact neighborhood ofinto an N-dimensional manifoldThen for any mapping that is sufficiently close to

14. DERIVATIONS We choose a basisfor Lemma 3 follows from classical results about the degree of mappings on spheres. To prove Theorem 2 we will first construct then apply Lemma 3 to a map and define Defineby

15. DERIVATIONS To show that H where we define the binary operation We observe that is nonsingular. We construct by satisfies the hypothesis of Lemma 3 it suffices, by the implicit function theorem, to prove

16. DERIVATIONS thus A direct computation shows that Furthermore, Lemma 2 and (2.5) imply that andare isomorphic topological groups. Nonsingularity follows since