Construction and calculation of reproducing kernel determined by various linear differential operators

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Abstract

We presented a method to construct and calculate the reproducing kernel for the linear differential operator with constant coefficients and a single latent root; further, we gave the formula for calculation. Additionally, by studying the recurrence relation of the reproducing kernel with arithmetic latent roots, we found that the reproducing kernel with a multi-knots interpolation constraint can be concisely represented by one with an initial-value constraint.

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 zi(t)=eλit(i=1,2,,m) 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) areM=111λ1λ2λmλ1m-1λ2m-1λmm-1andW(t)=Meλ1teλmt.That 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),G(t,s)=g(t,s)-1imφi(t)g(ti,s)=g(t,s)-1imφi(t)1jmbmjeλj(ti-s)(ti-s)+0,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

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