Construction and calculation of reproducing kernel determined by various linear differential operators
Section snippets
Introduction and preliminaries
A reproducing kernel is a basic tool for studying the spline interpolation of differential operators and also an important way to exactly determine the solution of an integral (differential) equation or to approximately solve such an equation [3], [9], [10]. Let Hm denote the function space on a finite interval, [0, T]. Hm = {f(t), t ∈ [0, T]: f(m−1)(t)} is absolutely continuous, and this space becomes a reproducing kernel Hilbert space (RKHS) if we endow it with some inner product. There are two
Case of initial-value constraint
Let the differential operator L be defined as in (9) and λ1, … , λm be a set of distinct numbers, either real or complex. Then is a basis system ofH1. Let li (1 ⩽ i ⩽ m) be the initial-constraint functional and satisfy lif = f(i−1)(0), f ∈ Hm. Then we know that the matrix M in Lemma 2 and the matrix W(t) in (6) areThat is to say, M is just the VanderMonde matrix. If we use vij to denote the element of M−1, at the ith row and the jth
Case of multi-knots interpolation constraint
Let the differential operator L be of the form in (9). And let li be the interpolation functional and satisfy li f = f(ti), i = 1, 2, … , m. Under this condition, we know that g(t, s) is still of the form in (14). From (8),where bmj is of the form in (13). Theorem 5 Let the differential operator L be of the form in (9) and let li be the interpolation functional and satisfy li f = f(ti), 0 ⩽ t1 < t2 < ⋯ < tm ⩽ T. Then the subspace H2 of Hm has a
References (10)
- et al.
Computing reproducing kernels with arbitrary boundary constraints [J]
SIAM J. Sci. Comput.
(1993) - et al.
The uniformity of spline interpolating operators and the best operators of interpolating approximation in spaces [J]
Math. Numer. Sin.
(2001) Spline interpolating operators and the best approximation of linear functionals in spaces [J]
Math. Numer. Sin.
(2002)- et al.
Reproducing Kernel Space Numerical Analysis[M]
(2004) - et al.
The exact solution of a kind integro-differential equation in space of reproducing kernel [J]
Math. Numer. Sin.
(1999)