Polynomial Splines Of Order Research Articles

Non-symmetric approximations of functional classes by splines on the real line

Let $S_{h,m}$, $h>0$, $m\in {\mathbb N}$, be the spaces of polynomial splines of order $m$ of deficiency 1 with nodes at the points $kh$, $k\in {\mathbb Z}$.
 We obtain exact values of the best $(\alpha, \beta)$-approximations by spaces $S_{h,m}\cap L_1({\mathbb R})$ in the space $L_1({\mathbb R})$ for the classes $W^r_{1,1}({\mathbb R})$, $r\in {\mathbb N}$, of functions, defined on the whole real line, integrable on ${\mathbb R}$ and such that their $r$th derivatives belong to the unit ball of $L_1({\mathbb R})$.
 These results generalize the well-known G.G. Magaril-Ilyaev's and V.M. Tikhomirov's results on the exact values of the best approximations of classes $W^r_{1,1}({\mathbb R})$ by splines from $S_{h,m}\cap L_1({\mathbb R})$ (case $\alpha=\beta=1$), as well as are non-periodic analogs of the V.F. Babenko's result on the best non-symmetric approximations of classes $W^r_1({\mathbb T})$ of $2\pi$-periodic functions with $r$th derivative belonging to the unit ball of $L_1({\mathbb T})$ by periodic polynomial splines of minimal deficiency.
 As a corollary of the main result, we obtain exact values of the best one-sided approximations of classes $W^r_1$ by polynomial splines from $S_{h,m}({\mathbb T})$. This result is a periodic analogue of the results of A.A. Ligun and V.G. Doronin on the best one-sided approximations of classes $W^r_1$ by spaces $S_{h,m}({\mathbb T})$.

On Error Estimates of Local Approximation by Splines

We consider the so-called simplest formula for local approximation by polynomial splines of order $ n $ (Schoenberg splines). The spline itself and all derivatives except that of the highest order, approximate a given function and its corresponding derivatives with the second order. We show that the jump of the highest derivative of order $ n-1 $ ; i. e., the value of discontinuity, divided by the meshsize, approximates the $ n $ th derivative of the original function. We found an asymptotic expansion of the jump.

A simple and general approach to fitting the discount curve under no-arbitrage constraints

We suggest a simple and general approach to fitting the discount curve under no-arbitrage constraints based on a penalized shape-constrained B-spline. The approach accommodates B-splines of any order and fitting both under the L1 and the L2 loss functions. An application to US STRIPS data from 2001–2015 suggests that polynomial splines of order three and four are mandatory to obtain reasonable fits. The choice of the loss function appears to be less relevant.

A tool path generation and contour error estimation method for four-axis serial machines

Although four-axis machines are widely used in welding, carving, and cutting revolving parts, much fewer four-axis tool path generation and contour error estimation researches have been studied when compared with five-axis ones. This paper proposes an off-line tool path generation and an on-line contour error estimation method for four-axis serial machines that are used for welding spatial intersecting curves on revolving parts. First, the kinematics module is derived, followed by proposing the tool path generation algorithm. The tool tip positions are represented by using a third order B-spline, while the tool orientations are represented by a third order polynomial-spline. Both splines, which form the C2 continuous tool path, are fitted to the curve length parameter of the tool tip positions. When compared with existing multi-axis trajectory generation methods, the proposed method is greatly simplified by mapping the tool pose commands from the Cartesian coordinate frame to the cylindrical coordinate frame. In the proposed contour error estimation method, the tool tip position contour error is estimated at first, followed by calculating the relative tool orientation contour error which is synchronized with the tool tip position contour error to one same pose on the desired trajectory. The proposed tool path generation and contour error estimation methods are experimentally verified on a four-axis welding machine controlled by an in-house developed CNC system. Experiment results show that the desired trajectory can be tracked continuously and steadily. The contour error can be estimated with high accuracy and reduced by 50% after performing on-line compensation.

Variable degree polynomial splines are Chebyshev splines

Variable degree polynomial (VDP) splines have recently proved themselves as a valuable tool in obtaining shape preserving approximations. However, some usual properties which one would expect of a spline space in order to be useful in geometric modeling, do not follow easily from their definition. This includes total positivity (TP) and variation diminishing, but also constructive algorithms based on knot insertion. We consider variable degree polynomial splines of order $k\geqslant 2$ spanned by $\{ 1,x,\ldots x^{k-3},(x-x_i)^{m_i-1},(x_{i+1}-x)^{n_i-1} \}$ on each subinterval $[x_i,x_{i+1}\rangle\subset [0,1]$ , iÂ?=Â?0,1, ...l. Most of the paper deals with non-polynomial case m i ,n i Â?Â?Â?[4,Â?Â?Â?), and polynomial splines known as VDP---splines are the special case when m i , n i are integers. We describe VDP---splines as being piecewisely spanned by a Canonical Complete Chebyshev system of functions whose measure vector is determined by positive rational functions p(x), q(x). These functions are such that variable degree splines belong piecewisely to the kernel of the differential operator $\frac{d}{dx} p \frac{d}{dx} q \frac{d^{k-2}} {dx^{k-2}}$ . Although the space of splines is not based on an Extended Chebyshev system, we argue that total positivity and variation diminishing still holds. Unlike the abstract results, constructive properties, like Marsden identity, recurrences for quasi-Bernstein polynomials and knot insertion algorithms may be more involved and we prove them only for VDP splines of orders 4 and 5.

On the exact values of the best approximations of classes of differentiable periodic functions by splines

We obtain the exact values of the best L1-approximations of classes WrF, r ∈ ℕ, of periodic functions whose rth derivative belongs to a given rearrangement-invariant set F, as well as of classes WrHω of periodic functions whose rth derivative has a given convex (upward) majorant ω(t) of the modulus of continuity, by subspaces of polynomial splines of order m ≥ r + 1 and of deficiency 1 with nodes at the points 2kπ/n and 2kπ/n + h, n ∈ ℕ, k ∈ ℤ, h ∈ (0, 2π/n). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

On one version of wavelet decompositions of spaces of polynomial splines

We study wavelet decompositions of the spaces of polynomial splines of order m on nonuniform grids, constructed via projections of Lagrange type. Bibliography: 3 titles.

Inequalities of the Bernstein type for splines of defect 2

We obtain new exact inequalities of the Bernstein type for periodic polynomial splines of order r and defect 2.

Exact values of best approximations for classes of periodic functions by splines of deficiency 2

We obtain exact values of best L1-approximations for the classes WrF, r ∈ ℕ, of periodic functions whose rth derivative belongs to a given rearrangement-invariant set F as well as for the classes WrHω of periodic functions whose rth derivative has a given convex (up) majorant ω(t) of the modulus of continuity by subspaces of polynomial splines of order m ≥ r + 1 of deficiency 2 with nodes at the points 2kπ/n, n ∈ ℕ, k ∈ ℤ. It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding function classes.

${ \mmb{\eta } }^{3}$-Splines for the Smooth Path Generation of Wheeled Mobile Robots

The paper deals with the generation of smooth paths for the inversion-based motion control of wheeled mobile robots. A new path primitive, called eta3-spline, is proposed. It is a seventh order polynomial spline which permits the interpolation of an arbitrary sequence of points with associated arbitrary tangent directions, curvatures, and curvature derivatives, so that an overall G3-path is planned. A G3-path or path with third order geometric continuity has continuous tangent vector, curvature, and curvature derivative along the arc length. Adopting this planning scheme and a dynamic path inversion technique, the robot's command velocities are continuous with continuous accelerations. The new primitive depends on a vector (eta) of six parameters that can be used to finely shape the path. The eta3-spline can generate or approximate, in a unified framework, a variety of curve primitives such as circular arcs, clothoids, spirals, etc. The paper includes theoretical results, path planning examples, and a note on general etak-splines.

FREQUENCY-DOMAIN IDENTIFICATION OF CONTINUOUS-TIME OUTPUT ERROR MODELS FROM NON-UNIFORMLY SAMPLED DATA

This paper treats the identification of continuous-time output error (OE) models based on sampled data. The exact method for doing this is well known both for data given in the time and frequency domains. The time domain approach however, becomes somewhat complex, especially for non-uniformly sampled data. In this paper we assume that the system input is a zero order hold signal and that the sampling rate is so high that for high frequencies the system behaves as a set of integrators. The conclusion is that if the system has relative degree ℓ then the output should be interpolated using an ℓ order polynomial spline function.

Impact of the shape of functions on the orders of piecewise polynomial and rational approximation

Let be the set of functions on a finite interval such that the divided differences of order of these functions are non-negative for all systems of distinct points . Let be the set of piecewise polynomial splines of order with free knots, and the set of rational functions , where and are polynomials of order . For the classes , where is the unit ball in , the precise orders and of the best approximations in the metrics are found for .

On the Exact Asymptotics of the Best Relative Approximations of Classes of Periodic Functions by Splines

We obtain the exact asymptotics (as n → ∞) of the best L1-approximations of classes \(W_1^r\) of periodic functions by splines s ∈ S2n, r − 1 and s ∈ S2n, r + k − 1 (S2n, r is the set of 2π-periodic polynomial splines of order r and defect 1 with nodes at the points kπ/n, k ∈ Z) under certain restrictions on their derivatives.

Stability of discrete orthogonal projections for continuous splines

In this paper Lp stability and convergence properties of discrete orthogonal projections on the finite element space Sh of continuous polynomial splines of order r are proved. The discrete inner products are defined by composite quadrature rules with positive weights on a sequence of nonuniform grids. It is assumed that the basic quadrature rule Q has at least r quadrature points in order to resolve Sh, but no accuracy is required. The main results are derived under minimal further assumptions, for example the rule Q is allowed to be non-symmetric, and no quasi-uniformity of the mesh is required. The corresponding stability of the orthogonal L2-projections has been studied by de Boor [1] and by Crouzeix and Thomee [2]. Stability of the first derivative of the projection is also proved, under an assumption (unless p = 1) of local quasi-uniformity of the mesh.

Polynomial spline signal approximations: filter design and asymptotic equivalence with Shannon's sampling theorem

The least-squares polynomial spline approximation of a signal g(t) in L/sub 2/(R) is obtained by projecting g(t) on S/sup n/(R) (the space of polynomial splines of order n). It is shown that this process can be linked to the classical problem of cardinal spline interpolation by first convolving g(t) with a B-spline of order n. More specifically, the coefficients of the B-spline interpolation of order 2n+1 of the sampled filtered sequence are identical to the coefficients of the least-squares approximation of g(t) of order n. It is shown that this approximation can be obtained from a succession of three basic operations: prefiltering, sampling, and postfiltering, which confirms the parallel with the classical sampling/reconstruction procedure for bandlimited signals. The frequency responses of these filters are determined for three equivalent spline representations using alternative sets of shift-invariant basis functions of S/sup n/(R): the standard expansion in terms of B-spline coefficients, a representation in terms of sampled signal values, and a representation using orthogonal basis functions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On best uniform approximations by splines in the presence of restrictions on their derivatives

Let Lp(l~p.~ 2) be the spaces of real-valued 2~-periodic functions with the corresponding norms ll'llp. By W~ (r = I, 2 .... ) we denote the class of functions ]~L= such that ](r-D(l(~ is locally absolutely continuous and [I](r)[Iv~i, while by $2~. r (n =1,2 .... ) we denote the set of 2~-periodic polynomial splines of order r, defect i, with nodes x k k ~ / n , k ~ Z If N CLp , then by the best approximation of the function l~Lv by the set N in the metric of Lp we mean the quantity

An independently working computer peripheral generating robot arm trajectories

An application specific peripheral processor is described, which independently carries out the task of generating the path for the desired motion of a manipulator. The trajectories are planned by the central computer in form of low order polynomial spline functions, whose parameters are then transferred to the peripheral. Here, the position, velocity, and acceleration values are constantly computed to be fed into the manipulator's control system. The algorithms employed by the peripheral and detailed below exploit various intrinsic properties of polynomials to minimise the arithmetic complexity. This distributed logic approach unburdens the central processor and utilises application specific inherent parallelism. Although a system's internal data transmission expense is drastically reduced, the concept allows much higher path update rates. The functional unit can be frequently applied and is thus easily standardisable.

L ∞ -Boundedness of L 2 -Projections on Splines for a Multiple Geometric Mesh

This paper concerns the L2-projectors from Loo to the normed linear space of polynomial splines. It is shown that for the multiple geometric meshes the Loo norms of the corresponding L2-projectors are bounded independently of the mesh ratio. 0. Introduction. Let us begin with some notation. Let k be a positive integer, and x = (x,) i z a real nondecreasing knot sequence with xi < XI?k, all i. Set x-00 : lim xi,I x:o := lim xji, I:= (x OO,x00). i-0 i o00o Fori E Z and x E I, define Mi k(X):= k[xi,..xi+k]( X)+i, Ni,k(X):= (Xi+k Xi)Mi,k(x)/k, where [po, . ., Pr]f denotes the rth divided difference of the function f at the points Po 0, Pr Then Ni k is called the i th B-spline of order k on the knot sequence x. For a E RZ, the rule f (x):= a(i)NiAk(x) defines a function on I if we take the sum to be pointwise. Every such function is called a polynomial spline of order k with the knot sequence x, and their collection is denoted by Skx* Further, let ? denote the normed linear space of bounded polynomial splines of order k with the knot sequence x and norm fll II:= sup I f(x) IX xEI We shall be concerned with P., the orthogonal projector onto ? with respect to the ordinary inner product (f, g) = ff(x)g(x) dx, but restricted to L,,(I). We want to bound its norm 11 P? II:= sup IIPsf II/1 III.o fEL,,(J) Received March 26, 1984; revised December 16, 1985. 1980 Mathematics Subject Classification. Primary 41A15, 15A60. *This is a part of the author's thesis, which was submitted to the graduate school, University of Wisconsin-Madison, in partial fullfillment of the requirement for Ph.D. degree. **Present address: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, The People's Republic of China. '1987 American Mathematical Society 0025-5718/87 $1.00 + $.25 per page 675 This content downloaded from 207.46.13.178 on Wed, 15 Jun 2016 07:00:20 UTC All use subject to http://about.jstor.org/terms

Further asymptotic properties of best approximation by splines

Let S n k, r denote the collection of polynomial splines of order k with at most ( n − 1) free knots, each of multiplicity r. This paper explicitly finds the constants C k, r, p so that lim n→∞ n k dist L p[0, 1] {ƒ, S n k,r} = C k,r,p ∥ƒ (k)∥ L σ[0,1] , where σ = p (kp + 1) and ƒ is sufficiently smooth. This completely fills the gap between previously known results for simple knots ( r = 1) and for piecewise polynomials ( r = k). We also consider similar asymptotic properties pertaining to approximation of vector functions by vector splines. We define the latter to be families of vector functions whose components are splines of order k with common knots.