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1.
2.
We study weighted approximation of multivariate functions for classes of standard and linear information in the worst case
and average case settings. Under natural assumptions, we show a relation between n th minimal errors for these two classes of information. This relation enables us to infer convergence and error bounds for
standard information, as well as the equivalence of tractability and strong tractability for the two classes.
April 11, 2001. Final version received: May 29, 2001. 相似文献
3.
An efficient algorithm for computing a smoothing polynomial splines under inequality constraints on derivatives is introduced where both order and breakpoints ofs can be prescribed arbitrarily. By using the B-spline representation ofs, the original semi-infinite constraints are replaced by stronger finite ones, leading to a least squares problem with linear inequality constraints. Then these constraints are transformed into simple box constraints by an appropriate substitution of variables so that efficient standard techniques for solving such problems can be applied. Moreover, the smoothing term commonly used is replaced by a cheaply computable approximation. All matrix transformations are realized by numerically stable Givens rotations, and the band structure of the problem is exploited as far as possible. 相似文献
4.
M. J. Johnson 《Constructive Approximation》1998,14(3):429-438
The functions φ
m
:=|.|
2m-d
if d is odd, and φ
m
:=|.|
2m-d
\log|.| if d is even, are known as surface splines, and are commonly used in the interpolation or approximation of smooth functions. We show that if one's domain is the unit
ball in R
d
, then the approximation order of the translates of φ
m
is at most m . This is in contrast to the case when the domain is all of R
d
where it is known that the approximation order is exactly 2m .
April 23, 1996. Date revised: May 5, 1997. 相似文献
5.
We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C1 cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods. 相似文献
6.
Ming-Jun Lai 《Numerical Algorithms》1992,2(1):33-38
We describe an algorithm to compute the B-nets of bivariate box splines on a three-or four-directional mesh. Two pseudo Fortran programs for those B-nets are given.Research supported by a Faculty Grant From the University of Utah Research Committee. 相似文献
7.
In this paper we deal with shape preserving interpolation of data sets given on rectangular grids. The aim is to show that there exist spline interpolants of the continuity classC
2 which areS-convex, monotone, or positive if the data sets have these properties. This is done by using particular rational bicubic splines defined on the grids introduced by the data. Interpolants of the desired type can be constructed by a simple search procedure. 相似文献
8.
Summary We prove convergence and error estimates in Sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus. Examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus-shaped domains in three or more dimensions. For odd-degree splines, we prove convergence of nodal collocation for any strongly elliptic operator. For even-degree splines and midpoint collocation, we find an additional condition for the convergence which is satisfied for the classical boundary integral operators. Our analysis is a generalization to higher dimensions of the corresponding analysis of Arnold and Wendland [4]. 相似文献
9.
In this paper we study the approximation power, the existence of a normalized B-basis and the structure of a degree-raising process for spaces of the formrequiring suitable assumptions on the functions u and v. The results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of CAGD. 相似文献
10.
Summary.
We describe algorithms for constructing point sets at which interpolation by
spaces of bivariate splines of arbitrary degree and smoothness is
possible. The splines are defined on rectangular partitions adding
one or two diagonals to each rectangle. The interpolation sets
are selected in such a way that the grid points of the partition
are contained in these sets, and no large linear systems have to be solved.
Our method is to generate a net of line segments and to choose point sets in
these segments which satisfy the Schoenberg-Whitney condition for
certain univariate spline spaces such that a principle of degree
reduction can be applied. In order to include the grid points in the
interpolation sets, we give a sufficient Schoenberg-Whitney type
condition for interpolation by bivariate splines supported in certain cones.
This approach is completely different
from the known interpolation methods for bivariate splines of degree at most
three. Our method is illustrated by some numerical examples.
Received
October 5, 1992 / Revised version received May 13, 1994 相似文献
11.
In this paper local bivariate C1 spline quasi-interpolants on a criss-cross triangulation of bounded rectangular domains are considered and a computational procedure for their construction is proposed. Numerical and graphical tests are provided. 相似文献
12.
Michael J. Johnson 《Numerische Mathematik》2000,84(3):451-474
Summary. We show that the -norm of the error in thin-plate spline interpolation in the unit disc decays like , where , under the assumptions that the function to be approximated is and that the interpolation points contain the finite grid .
Received February 13, 1998 / Published online September 24, 1999 相似文献
13.
A method is presented for the construction of positive rational splines of continuity classC
2. 相似文献
14.
J. M. Peña 《Numerische Mathematik》2006,103(1):151-154
This note is concerned with the characterizations and uniqueness of bases of finite dimensional spaces of univariate continuous
functions which are optimally stable for evaluation with respect to bases whose elements have no sign changes. 相似文献
15.
In convex interpolation the curvature of the interpolants should be as small as possible. We attack this problem by treating
interpolation subject to bounds on the curvature. In view of the concexity the lower bound is equal to zero while the upper
bound is assumed to be piecewise constant. The upper bounds are called fair with respect to a function class if the interpolation
problem becomes solvable for all data sets in strictly convex position. We derive fair a priori bounds for classes of quadraticC
1, cubicC
2, and quarticC
3 splines on refined grids. 相似文献
16.
Extending a recent result of Ulrich Reif on cardinal polynomial B-splines, we show that the cardinal Chebyshev B-spline basis
associated with a linear differential operator with constant real coefficients is orthonormal with respect to a unique weighted
Sobolev-type inner product. 相似文献
17.
This paper studies locally supported piecewise linear prewavelets on bounded triangulations of arbitrary topology. It is shown that a concrete choice of prewavelets form a basis of the wavelet space when the degree of the vertices in the triangulation is not too high. Received December 29, 1997 / Revised version received April 14, 1998 相似文献
18.
J.M. Peña 《Numerische Mathematik》2002,91(2):305-318
Summary. This paper is concerned with bases of finite dimensional spaces of univariate continuous functions which are optimally stable
for evaluation. The only bases considered are those whose elements have no sign changes. Among these, an optimally stable
basis is characterized under the assumption that the set of points where each basis function is nonzero is an interval. A
uniqueness result and many examples of such optimally stable bases are also provided.
Received May 26, 2000 / Published online August 17, 2001 相似文献
19.
Summary The quartic periodic and nonperiodic X-spline are separated from the class of all piecewise-quartic interpolatory polynomials and their orders of convergence, smoothness and complexity of construction are examined. In particular, error estimates of interpolation of smooth functions at uniformly spaced knots by eight quartic X-splines of special interest are presented. The results are illustrated by a numerical example. 相似文献
20.
Summary We discuss the problem of approximating a functionf of the radial distancer in
d
on 0r< by a spline function of degreem withn (variable) knots. The spline is to be constructed so as to match the first 2n moments off. We show that if a solution exists, it can be obtained from ann-point Gauss-Christoffel quadrature formula relative to an appropriate moment functional or, iff is suitably restricted, relative to a measure, both depending onf. The moment functional and the measure may or may not be positive definite. Pointwise convergence is discussed asn. Examples are given including distributions from statistical mechanics.The work of the first author was supported in part by the National Science Foundation under grant DCR-8320561. 相似文献