The Fourier transforms of B-splines with multiple integer knots are shown to satisfy a simple recursion relation. This recursion formula is applied to derive a generalized two–scale relation for B-splines with multiple knots. Furthermore, the structure of the corresponding autocorrelation symbol is investigated. In particular, it can be observed that the solvability of the cardinal Hermite spli...

Almost sure bounds are established on the uniform error of smoothing spline estimators in nonparametric regression with random designs. Some results of Einmahl and Mason (2005) are used to derive uniform error bounds for the approximation of the spline smoother by an “equivalent” reproducing kernel regression estimator, as well as for proving uniform error bounds on the reproducing kernel regre...

I discuss the production of low rank smoothers for d ≥ 1 dimensional data, which can be fitted by regression or penalized regression methods. The smoothers are constructed by a simple transformation and truncation of the basis that arises from the solution of the thinplate spline smoothing problem, and are optimal in the sense that the truncation is designed to result in the minimum possible pe...

This paper derives strong relations that boundary curves of a smooth complex of patches have to obey when the patches are computed by local averaging. These relations restrict the choice of reparameterizations for geometric continuity. In particular, when one bicubic tensor-product B-spline patch is associated with each facet of a quadrilateral mesh with n-valent vertices and we do not want seg...

Chaiken's algorithm is a procedure for inserting new knots into uniform quadratic Bspline curves by doubling the control points and taking two successive averages. Lane and Riesenfeld showed that Chaiken's algorithm extends to uniform B-spline curves of arbitrary degree. By generalizing the notion of successive averaging, we further extend Chaiken's algorithm to B-spline curves of arbitrary deg...

Curve fitting with piecewise splines for rapidly varying data has been a difficult problem. Often, the curve exhibits unwanted “wiggles” around those data points whose locations change rapidly in comparison to their neighboring data points. Many methods were proposed to solve this problem using non-homogeneous tension spline or fairing process. In this paper, a new approach using the concept of...