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INTERPOLATORY SOLUTIONS OF LINEAR ODE’S AND EXTENSIONS Wayne M. Lawton Dept. of Mathematics, National University of Singapore 2 Science Drive 2, Singapore 117543 [email protected] Yongwimon Lenbury Dept. of Mathematics, Mahidol University Rama 6 Road, Bangkok, Thailand 10400 SCOPE I G G its Lie algebra, identified with TI (G) is a connected Lie group with identity We examine the following initial value problem U CU; U(0) I Continuous solution Measure U : R G C : R G ISSUES Continuous dependence of solutions U (C ) Approximation & interpolation of continuous U : R G; U(0) I ~ ~ ~ by solutions U (C ), C C functions where M of ~ C M is a dense subspace of the space G-valued measures that vanish on finite sets Applications and extensions PRELIMINARIES Choose a euclidean structure , : G G R with norm | | : G R and let : G G R be the geodesic distance function defined by the induced right-invariant riemannian metric PRELIMINARIES M space of - valued measures on R without G point masses whose topology is given by seminorms || C ||k 0 | C(t) | dt, k 0. k P topological group of continuous G - valued functions W on R that satisfy W(0) I, equipped with the topology of uniform convergence over compact intervals, under pointwise multiplication functions having bounded variation locally BP PRELIMINARIES Lemma 1 A function U : R G 1 U is in M, CU is in B then (1.3) L(U)(t) 0 | C(s) | ds if and only if t gives the distance along the trajectory in G and (1.4) ρ(U(t),I) L(U)(t), t R PRELIMINARIES exp : G G (1.5) d dt exponential function exp(tX) X exp(tX), t R , X G S M subspace of step functions 0 : S B map control measures to solutions 0 (S) contains dense subset of interpolation set (1.7) B(t 1 ,.., t n , g1 ,.., g n ) B P(t1 ,.., t n , g1 ,.., g n ) 0 t 1 t n R , g1 ,..., g n G RESULT S M 0 extends to a continuous : M B that is one-to-one and onto. Furthermore, B is a subgroup of P and it forms topological Theorem 1 is dense and groups under both the topology of uniform convergence over compact intervals and the finer topology that makes the function a homeomorphism. DERIVATIONS , : G G G for matrix groups X , Y XY YX Lie bracket Ad : G Hom ( G,G) -1 Ad(g)(X) gXg Adjoint representation for matrix groups d ( 2.1) dt -1 Ad (U(t)) ( X ) [U U , Ad (U(t)) ( X ) ] 0 such that ( 2.2) | [X, Y] | | X | | Y | We choose DERIVATIONS The proof of Theorem 1 is based on the following Lemma 2 If 1 satisfy and U j B, C j U j U j , j 1,2,3,4 (2.3) U 3 U1U 2 1 (2.4) C4 C1 C2 , then (2.5) and C3 C1 Ad (U 3 ) C 2 , (2.6) L(U3 )(t) L(U4 )(t) | C 2 (s) | L(U4 )(s) K(s, t)ds t 0 where K(s, t) e ( L(U2 )(t) - L(U2 )(s)) Proof Apply Gronwall’s inequality to the following t L (U )(t) 0 | C (s) | ds 3 t3 3 0 | C (s) - Ad(U (s))(C (s)) | ds 1 3 2 t 0 ( | C (s) - C (s) | | C (s) - Ad(U (s))(C (s)) | ) d 1 2 2 3 2 t L(U )(t) 0 | C (s) - Ad(U (s))(C (s)) | ds 4 2 3 2 t s d L(U )(t) 0 | 0 dv Ad(U ( v ))(C ( v ))dv | ds 4 3 2 t s L(U )(t) 0 | 0 [ C ( v ), Ad(U ( v ))(C ( v ))]dv | ds 4 t s )(t) 0 0 | L(U 4 3 C (v) 3 | exp( ( L ( U 3 )( s ) L ( U 1 4 )( s )) 1 t )(t) 0 L(U L(U 2 )(s)f(s)ds 3 | C (v) | 2 dv ds RESULT Theorem 2 Let C M be a dense subspace. , Then for every positive integer n and pair of sequences 0 t 1 t n R , g1 ,..., g n G (C) contains a dense subset of P(t1 ,.., t n , g1 ,.., g n ). DERIVATIONS C B(t 1 ,.., t n , g1 ,.., g n ). It suffices to approximate C by elements in ( C) B(t 1,.., t n , g1,.., g n ). Lemma 3 Let f : D M be a homeomorphism m of a compact neighborhood of 0 R into an N-dimensional manifold M. Then for any mapping h : D M that is sufficiently close to f, f(0) h(D). Choose any DERIVATIONS 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 m n Define n by H:R G : M G (C) ((C)(t1 ), , (C)(tn )) We choose a basis B1 , , B d for G and define X i ( B1 i , , B d i ), i 1, , n; i ([t i-1 , t i )) H ( v) (C n v i Ad (U) (X i ) ), 1 d n v (v1 , , v d ) (R ) R m DERIVATIONS We observe that H(0) (g1 , , g n ). To show that H satisfies the hypothesis of Lemma 3 it suffices, by d H ( v) | v0 the implicit function theorem, to prove dv m n by is nonsingular. We construct F: R G F(v) (C (v1 X 1 ) (vn X n ) ). where we define the binary operation C1 C2 C1 Ad ((C1 )) (C2 ) DERIVATIONS A direct computation shows that d d H ( v) | v0 F ( v) | v0 dv dv Furthermore, Lemma 2 and (2.5) imply that thus (C1 C2 ) (C1 )(C2 ) M and B are isomorphic topological groups. Nonsingularity follows since F(v) (g1e1,..., gn e1...en ), ei exp((t i - t i-1 )X i vi ), i 1,..., n.