### A New Error Bound for Shifted Surface Spline Interpolation

*Lin-Tian Luh*

#### Abstract

Shifted surface spline is a frequently used radial function for scattered data interpolation. The most frequently used error bounds for this radial function are the one raised by Wu and Schaback in [17] and the one raised byMadych and Nelson in [14]. Both are O(dl) as d → 0, where l is a positive integer and d is the well-known fill-distance which roughly speaking measures the spacing of the data points. Then RBF people found that there should be an error bound of the form O(ω 1 d ) because shifted surface spline is smooth and every smooth function shares this property. This only problem was that the value of the cucial constant ω was unknown. Recently Luh raised an exponential-type error bound with convergence rate O(ω 1 d ) as d → 0 where 0 < ω < 1 is a fixed constant which can be accurately computed [11]. Although the exponential-type error bound converges much faster than the algebraic-type error bound, the constant ω is intensely influenced by the dimension n in the sense ω → 1 rapidly as n → ∞. Here the variable x of both the interpolated and interpolating functions lies in Rn. In this paper we present an error bound which is O(√dω′1d ) where 0 < ω′ < 1 is a fixed constant for any fixed n, and is only mildly influenced by n. In other words, ω′ → 1 very slowly as n → ∞, and ω′

DOI:

http://dx.doi.org/10.3968/j.sms.1923845220120101.001
### Refbacks

- There are currently no refbacks.

**Share us to:*** *

**Reminder**

If you have already registered in Journal A and plan to submit article(s) to Journal B, please click the **CATEGORIES**, or **JOURNALS A-Z **on the right side of the "**HOME**".

We only use three mailboxes as follows to deal with issues about paper acceptance, payment and submission of electronic versions of our journals to databases:

**caooc@hotmail.com; sms@cscanada.net; sms@cscanada.org **

Articles published in **Studies in Mathematical Sciences**** **are licensed under Creative Commons Attribution 4.0 (CC-BY).

**STUDIES IN MATHEMATICAL SCIENCES**** **Editorial Office

**Address**: 9375 Rue de Roissy Brossard, Québec, J4X 3A1, Canada

**Telephone**: 1-514-558 6138

Http://www.cscanada.net

Http://www.cscanada.org

**E-mail**:caooc@hotmail.com

Copyright © 2010 **Canadian Research & Development Centre of Sciences and Cultures**