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1.
Gerlind Plonka 《Numerical Algorithms》1993,5(1):51-62
Generalized Hermite spline interpolation with periodic splines of defect 2 on an equidistant lattice is considered. Then the classic periodic Hermite spline interpolation with shifted interpolation nodes is obtained as a special case.By means of a new generalization of Euler-Frobenius polynomials the symbol of the considered interpolation problem is defined. Using this symbol, a simple representation of the fundamental splines can be given. Furthermore, an efficient algorithm for the computation of the Hermite spline interpolant is obtained, which is mainly based on the fast Fourier transform. 相似文献
2.
Any quadratic spline with coinciding interpolating points and breakpoints can be uniquely determined by one initial condition. We examine how to choose the initial value in positive interpolation by using a norm of the spline. Monotone interpolation is also considered. The results are illuminated by numerical examples and by an application. 相似文献
3.
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
4.
M.J.D. Powell 《Numerische Mathematik》1994,68(1):107-128
Summary.
Let be a function from to that has
square
integrable second derivatives and let be the thin plate spline
interpolant
to at the points in
. We seek
bounds on the error when is in the convex
hull of
the interpolation points or when is close to at least one of
the
interpolation points but need not be in the convex hull. We find, for example,
that, if is inside a triangle whose vertices are any three
of the
interpolation points, then is bounded above by a
multiple of ,
where is the length of the longest side of the triangle and
where the
multiplier is independent of the interpolation points. Further, if
is any
bounded set in that is not a subset of a single straight
line, then we
prove that a sequence of thin plate spline interpolants converges to
uniformly on . Specifically, we require , where
is now the least upper bound on the numbers and where ,
, is
the least Euclidean
distance from to an interpolation point. Our method of
analysis applies
integration by parts and the Cauchy--Schwarz inequality to the scalar product
between second derivatives that occurs in the variational calculation of thin
plate spline interpolation.
Received November 10, 1993 / Revised version received March 1994 相似文献
5.
We derive a complex line integral representation for the
ebyshev norm of periodic spline interpolation operators of odd degree on uniform lattices. Several generalizations are indicated. 相似文献
6.
Sigrid Fredenhagen Hans Joachim Oberle Gerhard Opfer 《Journal of Approximation Theory》1999,96(2):182
In this paper we derive necessary optimality conditions for an interpolating spline function which minimizes the Holladay approximation of the energy functional and which stays monotone if the given interpolation data are monotone. To this end optimal control theory for state-restricted optimal control problems is applied. The necessary conditions yield a complete characterization of the optimal spline. In the case of two or three interpolation knots, which we call thelocalcase, the optimality conditions are treated analytically. They reduce to polynomial equations which can very easily be solved numerically. These results are used for the construction of a numerical algorithm for the optimal monotone spline in the general (global) case via Newton's method. Here, the local optimal spline serves as a favourable initial estimation for the additional grid points of the optimal spline. Some numerical examples are presented which are constructed by FORTRAN and MATLAB programs. 相似文献
7.
We review and extend the analysis of band circulant matrices which occur in the periodic spline interpolation theory with equispaced knots. Explicit bounds on the inverse matrices are given. 相似文献
8.
We develop methods for constructing sets of points which admit Lagrange and Hermite type interpolation by spaces of bivariate splines on rectangular and triangular partitions which are uniform, in general. These sets are generated by building up a net of lines and by placing points on these lines which satisfy interlacing properties for univariate spline spaces. 相似文献
9.
10.
Summary. We investigate splines from a variational point of view, which have the following properties: (a) they interpolate given
data, (b) they stay nonnegative, when the data are positive, (c) for a given integer they minimize the functional for all nonnegative, interpolating . We extend known results for to larger , in particular to and we find general necessary conditions for solutions of this restricted minimization problem. These conditions imply that
solutions are splines in an augmented grid. In addition, we find that the solutions are in and consist of piecewise polynomials in with respect to the augmented grid. We find that for general, odd there will be no boundary arcs which means (nontrivial) subintervals in which the spline is identically zero. We show also
that the occurrence of a boundary arc in an interval between two neighboring knots prohibits the existence of any further
knot in that interval. For we show that between given neighboring interpolation knots, the augmented grid has at most two additional grid points. In
the case of two interpolation knots (the local problem) we develop polynomial equations for the additional grid points which
can be used directly for numerical computation. For the general (global) problem we propose an algorithm which is based on
a Newton iteration for the additional grid points and which uses the local spline data as an initial guess. There are extensions
to other types of constraints such as two-sided restrictions, also ones which vary from interval to interval. As an illustration
several numerical examples including graphs of splines manufactured by MATLAB- and FORTRAN-programs are given.
Received November 16, 1995 / Revised version received February 24, 1997 相似文献
11.
V. V. Bogdanov 《Siberian Advances in Mathematics》2012,22(3):153-160
We consider the problem of interpolation of a function under the condition of the preservation of the nature of its piecewise monotonicity. We give sufficient conditions for the comonotone interpolation by a classical cubic C 2-spline in the representation based on the expansion of its first derivative in a basis consisting of B-splines. These conditions allow to determine whether the soobtained spline is comonotone without solving the interpolation problem. 相似文献
12.
It is known from the discrete harmonic analysis that the interpolation problem with equidistant interpolation points has a
unique solution. If the right-hand sides in the interpolation problem are fixed, the spline depends on two parameters: the
spline order and the number of points located between neighboring interpolation points. We find explicit expressions for the
limits of interpolation spllines with respect to each parameter separately and show that both repeated limits exist. We also
prove that these repeated limits are equal and their value is an interpolation trigonometric polynomial. Bibliography: 10
titles. Illustrations: 2 figures. 相似文献
13.
Valery A. Zheludev 《分析论及其应用》1998,14(4):66-88
In this paper we consider polynomial splines S(x) with equidistant nodes which may grow as O (|x|s). We present an integral representation of such splines with a distribution kernel. This representation is related to the
Fourier integral of slowly growing functions. The part of the Fourier exponentials herewith play the so called exponential
splines by Schoenberg. The integral representation provides a flexible tool for dealing with the growing equidistant splines.
First, it allows us to construct a rich library of splines possessing the property that translations of any such spline form
a basis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations
form a biorthogonal basis. As examples we present solutions of the problems of projection of a growing function onto spline
spaces and of spline interpolation of a growing function. We derive formulas for approximate evaluation of splines projecting
a function onto the spline space and establish therewith exact estimations of the approximation errors. 相似文献
14.
杨松林 《高等学校计算数学学报》2005,27(1):1-6
The matrix valued rational interpolation is very useful in the partial realization problem and model reduction for all the linear system theory. Lagrange basic functions have been used in matrix valued rational interpolation. In this paper, according to the property of cardinal spline interpolation, we constructed a kind of spline type matrix valued rational interpolation, which based on cardinal spline. This spline type interpolation can avoid instability of high order polynomial interpolation and we obtained a useful formula. 相似文献
15.
散乱数据的多项式自然样条光顺与广义插值 总被引:2,自引:3,他引:2
由于理论与实践的重要性,在多元插值方面有相当多的工作,如[1]-[11]。目前以箱样条(box splines),光滑余因子与B网方法以及薄板样条与径函数(radial basis function)方法比较活跃。前者具有良好的性质和丰富的结构,很快成为一个活跃的研究方向,最近更在小波(wavelet)变换理论研究上发挥了作用。但是,它一般只处理规则分划的问题,不能做多元散乱数据的插值。 相似文献
16.
Wei-Xian HuangGuo-Jin Wang 《Applied mathematics and computation》2011,217(9):4644-4653
This paper presents a new weighted bivariate blending rational spline interpolation based on function values. This spline interpolation has the following advantages: firstly, it can modify the shape of the interpolating surface by changing the parameters under the condition that the values of the interpolating nodes are fixed; secondly, the interpolating function is C1-continuous for any positive parameters; thirdly, the interpolating function has a simple and explicit mathematical representation; fourthly, the interpolating function only depends on the values of the function being interpolated, so the computation is simple. In addition, this paper discusses some properties of the interpolating function, such as the bases of the interpolating function, the matrix representation, the bounded property, the error between the interpolating function and the function being interpolated. 相似文献
17.
M. Reimer 《Numerische Mathematik》1984,44(3):417-424
Summary For oddm, the error of them-th-degree spline interpolant of power growth on an equidistant grid is estimated. The method is based on a decomposition formula for the spline function, which locally can be represented as an interpolation polynomial of degreem which is corrected by an (m+1)-st.-order difference term.Dedicated to Prof. Dr. Karl Zeller on the occasion of his 60th birthday 相似文献
18.
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
19.
Chordal cubic spline interpolation is fourth-order accurate 总被引:1,自引:0,他引:1
** Email: michaelf{at}ifi.uio.no It is well known that complete cubic spline interpolation offunctions with four continuous derivatives is fourth-order accurate.In this paper we show that this kind of interpolation, whenused to construct parametric spline curves through sequencesof points in any space dimension, is again fourth-order accurateif the parameter intervals are chosen by chord length. We alsoshow how such chordal spline interpolants can be used to approximatethe arc-length derivatives of a curve and its length. 相似文献
20.
How to Get the Original Discrete Approximation for the Pyramid Algorithm of Multiresolution
Decomposition 总被引:2,自引:0,他引:2
§1.IntroductionInthispaper,weusethenotationsZ,R,L2(R)andl2forthesetofintegers,re-als,squareintegrablefunctionsandsquaresummab... 相似文献