| Let (OMEGA) be an open, bounded set in (//R)('n), and let A be a finite subset of (OMEGA). For f in H('k)((OMEGA)), where k > n/2, a spline s satisfying (-1)('k)(DELTA)('k)s(x) = 0 for x in (OMEGA)-A and solving the interpolation problem:; s(a) = f(a) a(epsilon)A; (f-s) (epsilon) H(,0)('k)((OMEGA)); is shown to exist and to exhibit many of the properties characteristic of single-variable polynomial spline interpolants. The proof which establishes existence provides insight into the construction of these splines. It is also extended to include splines which interpolate derivatives of the function f at the points of A.; For 1 (LESSTHEQ) p < (INFIN) and kp > n, the seminorms; (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI); 0 (LESSTHEQ) j (LESSTHEQ) k, are shown to satisfy the inequality; (VBAR)f(VBAR)(,j,p,(//R))n (LESSTHEQ) Ch('k-j)(VBAR)f(VBAR)(,k,p,(//R))n; for a constant C depending only on k, p, and n, whenever f is a function in H('k,p)((//R)('n)) whose set Y of zeros is such that; (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI); This inequality is used to obtain error estimates for polyharmonic spline interpolation which are analogous to error estimates for the single-variable generalized L-spline interpolation.; For k > n/2 and R > 0, the Green's functions for (-1)('k)(DELTA)('k) and (OMEGA)(,R) = {lcub}x(epsilon)(//R)('n): (VBAR)x(VBAR) < R{rcub} are derived and are used to obtain examples of spline interpolants for three functions f:(//R)('2) (--->) (//R)('1). |
