Mixed Interpolating-Smoothing Splines and the V-Spline

Document Type

Article

Publication Date

10-1-2006

Publication Title

Journal of Mathematical Analysis and Applications

DOI

10.1016/j.jmaa.2005.07.007

ISSN

0022-247X

Abstract

In their monograph, Bezhaev and Vasilenko have characterized the “mixed interpolating–smoothing spline” in the abstract setting of a Hilbert space. In this paper, we derive a similar characterization under slightly more general conditions. This is specialized to the finite-dimensional case, and applied to a few well-known problems, including the ν-spline (a piecewise polynomial spline in tension) and near-interpolation, as well as interpolation and smoothing. In particular, one of the main objectives in this paper is to show that the ν-spline is actually a mixed spline, an observation that we believe was not known prior to this work. We also show that the ν-spline is a limiting case of smoothing splines as certain weights increase to infinity, and a limiting case of near-interpolants as certain tolerances decrease to zero. We conclude with an iteration used to construct curvature-bounded ν-spline curves.

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