Multivariate polyharmonic spline interpolation

Thumbnail Image
Date
1981
Authors
Potter, Evelyn
Major Professor
Advisor
Committee Member
Journal Title
Journal ISSN
Volume Title
Publisher
Altmetrics
Authors
Research Projects
Organizational Units
Organizational Unit
Journal Issue
Is Version Of
Versions
Series
Department
Mathematics
Abstract

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 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) = x(epsilon)(//R)('n): (VBAR)x(VBAR) ) (//R)('1).

Comments
Description
Keywords
Citation
Source
Subject Categories
Copyright
Thu Jan 01 00:00:00 UTC 1981