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1.
Macro-elements of arbitrary smoothness are constructed on Powell-Sabin triangle splits. These elements are useful for solving boundary-value problems and for interpolation of Hermite data. It is shown that they are optimal with respect to spline degree, and we believe they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for certain superspline spaces defined over Powell-Sabin refinements. These bases are shown to be stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have optimal order approximation power.
2.
Summary. Macro-elements of arbitrary smoothness are constructed on Clough-Tocher triangle splits. These elements can be used for solving
boundary-value problems or for interpolation of Hermite data, and are shown to be optimal with respect to spline degree. We
conjecture they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for
certain superspline spaces defined over Clough-Tocher refinements of arbitrary triangulations. These bases are shown to be
stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have
optimal order approximation power.
Received November 18, 1999 / Published online October 16, 2000 相似文献
3.
Upper and lower bounds are provided on the dimension of bivariate polynomial superspline spaces which are defined by enforcing
smoothness conditions across the interior edges of the underlying triangulation. The results generalize known bounds for classical
spline spaces. As an example of the usefulness of such bounds, we show how they can be applied to analyze a new macroelement. 相似文献
4.
Marcel Rosenthal 《Mathematische Nachrichten》2013,286(1):59-87
We consider local means with bounded smoothness for Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces. Based on those we derive characterizations of these spaces in terms of Daubechies, Meyer, Bernstein (spline) and more general r‐regular (father) wavelets, finally in terms of (biorthogonal) wavelets which can serve as molecules and local means, respectively. Hereby both, local means and wavelet decompositions satisfy natural conditions concerning smoothness and cancellation (moment conditions). Moreover, the given representations by wavelets are unique and yield isomorphisms between the considered function spaces and appropriate sequence spaces of wavelet coefficients. These wavelet representations lead to wavelet bases if, and only if, the function spaces coincide with certain classical Besov‐Triebel‐Lizorkin spaces. 相似文献
5.
Zhi-qiangXu Ren-hongWang 《计算数学(英文版)》2004,22(6):807-816
Super splines are bivariate splines defined on triangulations, where the smoothness enforced at the vertices is larger than the smoothness enforced across the edges. In this paper, the smoothness conditions and conformality conditions for super splines are presented. Three locally supported super splines on type-1 triangulation are presented. Moreover, the criteria to select local bases is also givenBy using local supported super spline function, a variation-diminishing operator is built. The approximation properties of the operator are also presented. 相似文献
6.
In this paper, the dimension of the spaces of bivariate spline with degree less that 2r and smoothness order r on the Morgan-Scott triangulation is considered. The concept of the instability degree in the dimension of spaces of bivariate spline is presented. The results in the paper make us conjecture the instability degree in the dimension of spaces of bivariate spline is infinity. 相似文献
7.
Marie-Laurence Mazure 《Numerische Mathematik》2011,119(3):517-556
In the present work we determine all Chebyshevian spline spaces good for geometric design. By Chebyshevian spline space we
mean a space of splines with sections in different Extended Chebyshev spaces and with connection matrices at the knots. We
say that such a spline space is good for design when it possesses blossoms. To justify the terminology, let us recall that,
in this general framework, existence of blossoms (defined on a restricted set of tuples) makes it possible to develop all
the classical geometric design algorithms for splines. Furthermore, existence of blossoms is equivalent to existence of a
B-spline bases both in the spline space itself and in all other spline spaces derived from it by insertion of knots. We show
that Chebyshevian spline spaces good for design can be described by linear piecewise differential operators associated with
systems of piecewise weight functions, with respect to which the connection matrices are identity matrices. Many interesting
consequences can be drawn from the latter characterisation: as an example, all Chebsyhevian spline spaces good for design
can be built by means of integral recurrence relations. 相似文献
8.
《Journal of Computational and Applied Mathematics》2012,236(5):765-774
In this paper, the dimensions of bivariate spline spaces are studied using the Smoothing Cofactor-Conformality method. Based on the analysis on the conformality condition at one interior vertex, the stability (or singularity to the contrary) of the dimensions of general spline spaces is discussed in detail. By the aid of directed partition some new results on dimensions are obtained with the corresponding constraints depending on the degree, the smoothness order of the spline spaces and the structure of the partition as well. 相似文献
9.
本文首先研究了Υ阶广义样条类在Orlicz空间内的极值问题,由此进一步考虑了光滑函数类Ω_∞~Υ[0,1]在Orlicz空间内的n宽度的精确估计问题.最后还讨论了相应的对偶情形. 相似文献
10.
In this paper, the dimensions of bivariate spline spaces are studied using the Smoothing Cofactor-Conformality method. Based on the analysis on the conformality condition at one interior vertex, the stability (or singularity to the contrary) of the dimensions of general spline spaces is discussed in detail. By the aid of directed partition some new results on dimensions are obtained with the corresponding constraints depending on the degree, the smoothness order of the spline spaces and the structure of the partition as well. 相似文献
11.
12.
We consider the numerical solution of one-dimensional Fredholmintegral equations of the second kind by the Galerkin and collocationmethods and their iterated variants, using spline bases. Inparticular, we state and prove new superconvergence resultsfor the iterated solutions, under general smoothness requirementson the kernel and solution. We find that the smoothness requirementsfor the iterated collocation method are more stringent thanthose for the iterated Galerkin method, and show by examplethat these more stringent smoothness conditions are in a certainsense necessary. In the light of these results the Galerkinand collocation schernes are compared. 相似文献
13.
It is well-known that the basic properties of a bivariate spline space such as dimension and approximation order depend on the geometric structure of the partition. The dependence of geometric structure results in the fact that the dimension of a C
1 cubic spline space over an arbitrary triangulation becomes a well-known open problem. In this paper, by employing a new group of smoothness conditions and conformality conditions, we determine the dimension of bivariate C
1 cubic spline spaces over a so-called even stratified triangulation. 相似文献
14.
A. Kamont 《Constructive Approximation》1996,12(1):111-125
This paper deals with the method of hyperbolic summation of tensor product orthogonal spline functions onI d. The spaces, defined in terms of the order of the best approximation by the elements of the space spanned by the tensor product functions with indices from a given hyperbolic set, are described both in terms of the coefficients in some basis and as interpolation spaces. Moreover, the hyperbolic modulus of smoothness is studied, and some relations between hyperbolic summation and hyperbolic modulus of smoothness are established. 相似文献
15.
Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines (or ponits), and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u+1^u (△MS^u) are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal's theorem is generalized to algebraic plane curves of degree n≥3. 相似文献
16.
17.
Yu. K. Dem’yanovich 《Journal of Mathematical Sciences》2010,170(4):444-495
We obtain sufficient conditions for the existence, smoothness, and embedding of spline spaces, construct a biorthogonal system
of functionals, find calibration relations, and study the asymptotic behavior of minimal first and second order splines of
Lagrange type. Bibliography: 9 titles. 相似文献
18.
Marie-Laurence Mazure 《Numerical Algorithms》2011,58(3):399-438
We prove that a general class of splines with sections in different Extended Chebyshev spaces or in different quasi Extended
Chebyshev spaces can be viewed as quasi Chebyshevian splines, that is, as splines with all sections in a single convenient
quasi Extended Chebyshev space. As a result, we can affirm the presence of blossoms in the corresponding spline spaces, with
all the important consequences inherent in blossoms, namely, the possibility of developing all design algorithms for splines,
the existence of B-splines bases, along with their optimality. 相似文献
19.
Rong-Qing Jia 《Advances in Computational Mathematics》2009,30(2):177-200
In this paper we investigate spline wavelets on the interval with homogeneous boundary conditions. Starting with a pair of
families of B-splines on the unit interval, we give a general method to explicitly construct wavelets satisfying the desired
homogeneous boundary conditions. On the basis of a new development of multiresolution analysis, we show that these wavelets
form Riesz bases of certain Sobolev spaces. The wavelet bases investigated in this paper are suitable for numerical solutions
of ordinary and partial differential equations.
Supported in part by NSERC Canada under Grant OGP 121336. 相似文献
20.
Which spaces for design? 总被引:1,自引:0,他引:1
Marie-Laurence Mazure 《Numerische Mathematik》2008,110(3):357-392
We determine the largest class of spaces of sufficient regularity which are suitable for design in the sense that they do
possess blossoms. It is the class of all spaces containing constants of which the spaces derived under differentiation are
Quasi Extended Chebyshev spaces, i.e., they permit Hermite interpolation, Taylor interpolation excepted. It is also the class of all spaces which possess
Bernstein bases, or of all spaces for which any associated spline space does possess a B-spline basis. Note that blossoms
guarantee that such bases are normalised totally positive bases. They even are the optimal ones. 相似文献