Piecewise linear prewavelets on arbitrary triangulations

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     

Cubic Spline Prewavelets on the Four-Directional Mesh

Dedicated to Professor M. J. D. Powell on the occasion of his sixty-fifth birthday and his retirement. In this paper, we design differentiable, two-dimensional, piecewise polynomial cubic prewavelets of particularly small compact support. They are given in closed form, and provide stable, orthogonal decompositions of L ² (R ² ) . In particular, the splines we use in our prewavelet constructions give rise to stable bases of spline spaces that contain all cubic polynomials, whereas the more familiar box spline constructions cannot reproduce all cubic polynomials, unless resorting to a box spline of higher polynomial degree.     

Algebraic analysis of multigrid algorithms

We study the convergence rate of multilevel algorithms from an algebraic point of view. This requires a detailed analysis of the constant in the strengthened Cauchy–Schwarz inequality between the coarse‐grid space and a so‐called complementary space. This complementary space may be spanned by standard hierarchical basis functions, prewavelets or generalized prewavelets. Using generalized prewavelets, we are able to derive a constant in the strengthened Cauchy–Schwarz inequality which is less than 0.31 for the L² and H¹ bilinear form. This implies a convergence rate less than 0.15. So, we are able to prove fast multilevel convergence. Furthermore, we obtain robust estimations of the convergence rate for a large class of anisotropic ellipic equations, even for some that are not H¹‐elliptic. Copyright © 1999 John Wiley & Sons, Ltd.     

Piecewise linear prewavelets over type-2 triangulations

In this article, we study the construction of piecewise linear prewavelets over type-2 triangulations. Different from a so-called semi-prewavelet approach, we investigate the orthogonal conditions directly and obtain parameterized prewavelets with a smaller support. The conditions for parameterized prewavelet basis on the parameters are also given.     

加密网格点二元局部基插值样条函数

1.简介 由于在理论以及应用两方面的重要性,多元样条引起了许多人的注意([6],[7]),紧支撑光滑分片多项式函数对于曲面的逼近是一个十分有效的工具。由于它们的局部支撑性,它们很容易求值;由于它们的光滑性,它们能被应用到要满足一定光滑条件的情况下;由于它们是紧支撑的,它们的线性包有很大的逼近灵活性,而且用它们构造逼近方法来解决的系统是     

多元弱样条空间和最小确定集

根据研究多元弱样条函数的B—网方法,给出了某些多元弱样条函数空间的最小确定集的构造方法,并从而求出了它们的维数．本文还讨论了对偶基的局部支集性质。     

A remark on translates of a box spline

We prove an express theorem on the stability of integer translates of a box spline and clear proposition 4. 10. 2 in [6] for box spline case. Also, we provide a way to construct a linear projection from L_p to the space of integer translates of a box spline. Consequently, the equivalence among the stability, global linear independence and local linear independence of translates of a box spile follows. Partially supported by ARO contract No. DAAL 4-87-0025     

Spline Split Quaternion Interpolation in Minkowski Space

Spherical spline quaternion interpolation has been done on sphere in Euclidean space using quaternions. In this paper, we have done the spline split quaternion interpolation on hyperbolic sphere in Minkowski space using split quaternions and metric Lorentz. This interpolation curve is called spherical spline split quaternion interpolation in Minkowski space (MSquad).     

Construction of multivariate compactly supported prewavelets in space and pre-Riesz bases in Sobolev spaces

We give a new constructive method for finding compactly supported prewavelets in L₂ spaces in the multivariate setting. This method works for any dimensional space. When this method is generalized to the Sobolev space setting, it produces a pre-Riesz basis for which can be useful for applications.     

How to build all Chebyshevian spline spaces good for geometric design?

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.     

Bivariate Polynomial Natural Spline Interpolation Algorithms with Local Basis for Scattered Data

Because of its importance in both theory and applications, multivariate splines have attracted special attention in many fields. Based on the theory of spline functions in Hilbert spaces, bivariate polynomial natural splines for interpolating, smoothing or generalized interpolating of scattered data over an arbitrary domain are constructed with one-sided functions. However, this method is not well suited for large scale numerical applications. In this paper, a new locally supported basis for the bivariate polynomial natural spline space is constructed. Some properties of this basis are also discussed. Methods to order scattered data are shown and algorithms for bivariate polynomial natural spline interpolating are constructed. The interpolating coefficient matrix is sparse, and thus, the algorithms can be easily implemented in a computer.     

Bell-shaped nonstationary refinable ripplets

We study the approximation properties of the class of nonstationary refinable ripplets introduced in Gori and Pitolli (2008). These functions are the solution of an infinite set of nonstationary refinable equations and are defined through sequences of scaling masks that have an explicit expression. Moreover, they are variation-diminishing and highly localized in the scale-time plane, properties that make them particularly attractive in applications. Here, we prove that they enjoy Strang-Fix conditions, convolution and differentiation rules and that they are bell-shaped. Then, we construct the corresponding minimally supported nonstationary prewavelets and give an iterative algorithm to evaluate the prewavelet masks. Finally, we give a procedure to construct the associated nonstationary biorthogonal bases and filters to be used in efficient decomposition and reconstruction algorithms. As an example, we calculate the prewavelet masks and the nonstationary biorthogonal filter pairs corresponding to the C ² nonstationary scaling functions in the class and construct the corresponding prewavelets and biorthogonal bases. A simple test showing their good performances in the analysis of a spike-like signal is also presented.     

Dimension elevation for Chebyshevian splines

Dimension elevation refers to the Chebyshevian version of the classical degree elevation process for polynomials or polynomial splines. In this paper, we consider the case of splines. The original spline space is based on a given Extended Chebsyhev space \mathbbE{\mathbb{E}} contained in another Extended Chebsyhev space \mathbbE^*{\mathbb{E}}^* of dimension increased by one. The original spline space, based on \mathbbE{\mathbb{E}}, is then embedded in a larger one, based on \mathbbE^*\mathbb{E}^*. Thanks to blossoms we show how to compute the new poles of any spline in the original spline space in terms of its initial poles.     

Dimensions of spline spaces over non-rectangular T-meshes

Spline spaces over rectangular T-meshes have been discussed in many papers. In this paper, we consider spline spaces over non-rectangular T-meshes. The dimension formulae of spline spaces over special simply connected T-meshes have been obtained. For T-meshes with holes, we discover a new type of dimension instability. We construct a relationship between the dimension of the spline space over a T-mesh \(\mathcal {T}\) with holes and the dimension of the spline space over a simply connected T-mesh associated with \(\mathcal {T}\).     

On the Problem of Instability in the Dimensions of Spline Spaces over T-Meshes with T-Cycles

The T-meshes are local modification of rectangular meshes which allow T-junctions. The splines over T-meshes are involved in many fields, such as finite element methods, CAGD etc. The dimension of a spline space is a basic problem for the theories and applications of splines. However, the problem of determining the dimension of a spline space is difficult since it heavily depends on the geometric properties of the partition. In many cases, the dimension is unstable. In this paper, we study the instability in the dimensions of spline spaces over T-meshes by using the smoothing cofactor-conformality method. The modified dimension formulas of spline spaces over T-meshes with T-cycles are also presented. Moreover, some examples are given to illustrate the instability in the dimensions of the spline spaces over some special meshes.     

On explicit basis of bivariate spline space

For a subdivision Δ of a region in d-dimensional Euclidean space, we consider computation of dimension and of basis function in spline space S _k ^r (Δ) consisting of all C piecewise polynomial functions over Δ of degree at most k. A computational scheme is presented for computing the dimension and bases of spline space S _k ^r (Δ). This scheme based on the Grobner basis algorithm and the smooth co-factor method for computing multivariate spline. For bivariate splines, explicit basis functions of S _k ^r (Δ) are obtained for any integer k and r when Δ is a cross-cut partition. The Project is partly supported by the Science and Technology New Star Plan of Beijing and Education Committee of Beijing.     

Integral representation of slowly growing equidistant splines

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.     

Least supported bases and local linear independence

Summary. We introduce the concept of least supported basis, which is very useful for numerical purposes. We prove that this concept is equivalent to the local linear independence of the basis. For any given locally linearly independent basis we characterize all the bases of the space sharing the same property. Several examples for spline spaces are given. Received December 4, 1992 / Revised version received March 2, 1993     

ANTIPERIODIC WAVELETS

1.PreliminaryAswekn0w,manyauth0rshavemadegreateff0rtsinconstructingthe0rthonormalorbiorthon0rmalbasis0nthewholereallinem10ronthewh0len-dimensionalspaceMnl"7],butinmanypracticalpr0blems0neneedst0construct0rthon0rmalbasisonsomefiniteintervawiths0meboundaryconditi0ns.HerewepresentamethodofconstructingtheantiperiodicorthonormalwaveletsbasisontheintervaI=[0,2x1.Maindifficultyintheaboveproblemisthec0nstructi0noftheorthonormalbasisofWm-l-theorth0g0nalc0mplement0fVm-1inVm-thekeystepisthatwehavetocon…     

Local Sampling and Reconstruction in Shift-Invariant Spaces and Their Applications in Spline Subspaces

The local reconstruction from samples is one of the most desirable properties for many applications in signal processing. Local sampling is practically useful since we need only to consider a signal on a bounded interval and computer can only process finite samples. However, the local sampling and reconstruction problem has not been given as much attention. Most of known results concern global sampling and reconstruction. There are only a few results about local sampling and reconstruction in spline subspaces. In this article, we study local sampling and reconstruction in general shift-invariant spaces and multiple generated shift-invariant spaces with compactly supported generators. Then we give several applications in spline subspaces and multiple generated spline subspaces.