and Applied Analysis 3 Proof. Let p(z) be given by (11), which clearly belongs to the class A(n). Now differentiating (11), we have p 󸀠 (z) = ( f (z) ∗ g (z) z ) 1/(p+b−1) × 1 p + b − 1 { z(f (z) ∗ g (z)) 󸀠 f (z) ∗ g (z) + b − 1} (20) which gives 󵄨󵄨󵄨󵄨 arg p (z)󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 arg( f (z) ∗ g (z) z ) 1/(p+b−1) + arg{ 1 p+b−1 ( z(f (z) ∗ g (z)) 󸀠 f (z) ∗ g (z) +b−1)} 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 . (21) Thus using (...

The algorithms and programs in this handy book deal almost exclusively with cubic splines. A short introduction stresses the maximum "smoothness" of "natural" splines. The next chapter reviews the numerical solution of tridiagonal linear systems by Gauss elimination and of block tridiagonal linear systems by relaxation. Chapter 3 offers programs for cubic spline interpolation with various (sepa...

We construct bootstrap conndence intervals for smoothing spline and smoothing spline ANOVA estimates based on Gaussian data, and penalized likelihood smoothing spline estimates based on data from exponential families. Several variations of bootstrap conndence intervals are considered and compared. We nd that the commonly used bootstrap percentile intervals are inferior to the T intervals and to...

This paper presents a new fast and local method of 3D surface reconstruction for scattered data. The algorithm makes use of quasiinterpolants to compute the control points from a coarse to fine hierarchy to generate a sequence of bicubic B-spline functions whose sum approaches to the desired interpolation function. Quasi-interpolants gives a procedure for deriving local spline approximation met...

In this paper we intend to generate some set of optimal trajectories according to the number of control points has been applied for parameterizing those using B-spline curves. The trajectories are used to generate an optimal locomotion gait in a crawling worm-like robot. Due to gait design considerations it is desired to minimize the required torques in a cycle of gait. Similar to caterpillars,...