spline space and its b splines on an n 1 direction mesh in r n it is well known that splines play an important role in many fields especially their close relationship with wavelets makes them have more widespread applications in numerous scientific and engineering domains univariate and bivariate splines have been well studied and lots of results have been obtained because of the intrinsic difficulty between bivariate case and higher dimension three or more dimensions settings the study of splines on higher dimensions are very limited for example the study of the bivariate splines on a three direction mesh triangulation has obtained many important and excellent results but almost all of those results have no analog generalization to higher dimensions in this paper we will study the higher dimension splines defined on n 1 mesh simplical partitions which is the analog of bivariate splines on three mesh triangulations we have also pointed out many interesting differences between bivariate splines and higher dimensional cases our main results are that similar to bivariate and trivariate cases a necessary and sufficient condition for skr to contain a b spline is k r 1 n 1 for r 1 being odd and k r n 1 1 for r 0 being even

a new formula for box splines on three directional meshes 1 introductionbox splines on three directional meshes and their translates provide a usefultool in the finite element method see e g 4 5 this class of multivariatesplines is contained in a broader class of splines introduced by de boor anddevore 1 some basic recurrence relations obeyed by the latter splines havebeen derived in 2 computations with these recursions could be cumbersomeand time consuming in this note we offer a closed formula for multivariatebox splines on three directional meshes see 3 in 2 we give notation and definitions which will be used in subsequent sec tions the main result is derived in 3 an application is given in 4 2 notation and definitionsby ej 1 j s we will denote the jth coordinate vector in k5 also let

on smooth multivariate spline functions in this paper the dimensions of bivariate spline spaces with simple cross cut grid partitions are determined and expressions of their basis functions are given consequently the closures of these spaces over all partitions of the same type can be determined a somewhat more detailed study on bivariate splines with rectangular grid partitions is included the results in this paper can be applied to problems on interpolation and approximation by bivariate spline functions

on the linear independence of multivariate b splines i triangulations of simploids this paper is concerned with the construction of certain collections of linearly independent multivariate b splines whose linear span is appropriate for approximation by constructing linear projectors onto these spline spaces we are able to estimate local and global approximation errors as well as the condition number of the b spline basis our approach is based upon some combinatorial properties of certain triangulations of a class of n polytopes which we call simploids

integral error formulae for the scale of mean value interpolations which includes kergin and hakopian interpolation in this paper we provide an integral error formula for a certain scale of mean value interpolations which includes the multivariate polynomial interpolation schemes of kergin and hakopian this formula involves only derivatives of order one higher than the degree of the interpolating polynomial space and from it we can obtain sharp l infty estimates these l infty estimates are precisely those that numerical analysts want to guarantee that a scheme based on such an interpolation has the maximum possible order

on the dimension of bivariate spline spaces of smoothnessr and degreed 3r 1 we consider the well known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally to date dimension formulae for such spaces have been established only whend 3r 2 except for the special case wherer 1 andd 4 in this paper we establish dimension formulae for allr 1 andd 3r 1 for almost all triangulations

attenuation factors in multivariate fourier analysis w gautschi s theory of attenuation factors for families of periodic functions in one variable is extended to families of functions in several variables again the linearity and the translation invariance of the operator which maps the data space onto the family are crucial special results are obtained for tensor product families and for the interpolation by translates of one generating function interesting examples are provided by box splines which include certain finite elements

the generic dimension of the space of c 1 splines of degree d 8 on tetrahedral decompositions this paper considers the linear space of globally differentiable piecewise polynomial functions defined on a three dimensional polyhedral domain which has been partitioned into tetrahedra combining bernstein bezier methods and combinatorial and geometric techniques from rigidity theory this paper gives an explicit expression for the generic dimension of this space for sufficiently large polynomial degrees d geq 8 this is the first general dimension statement of its kind

lyapunov exponent for the parabolic anderson model in rd abstract we consider the asymptotic almost sure behavior of the solution of the equation u t x u 0 x 2 0 t u s x d s 0 t u s x w x s where w x x r d is a field of brownian motions in fact we establish existence of the lyapunov exponent lim t 1 t log u t x we also show that c 1 1 3 c 2 1 5 as 0 under the assumption that the correlation function of the background field w x x r d is c for 1 2

symmetric box splines on root lattices root lattices are efficient sampling lattices for reconstructing isotropic signals in arbitrary dimensions due to their highly symmetric structure one root lattice the cartesian grid is almost exclusively used since it matches the coordinate grid but it is less efficient than other root lattices box splines on the other hand generalize tensor product b splines by allowing non cartesian directions they provide in any number of dimensions higher order reconstructions of fields often of higher efficiency than tensored b splines but on non cartesian lattices such as the bcc body centered cubic or the fcc face centered cubic lattice only some box splines and then only up to dimension three have been investigated this paper derives and completely characterizes efficient symmetric box spline reconstruction filters on all irreducible root lattices that exist in any number of dimensionsn 2 n 3 for d n and d n lattices in all cases box splines are constructed by convolution using the lattice directions generalizing the known constructions in two and three variables for each box spline we document the basic properties for computational use the polynomial degree the continuity the linear independence of shifts on the lattice and optimal quasi interpolants for fast approximation of fields

multivariate spline functions b spline basis and polynomial interpolations the main result of this paper is a formula relating a multivariate b spline and a certain directional derivative to a lower order b spline the structure of discontinuous b splines is given a formula relates the discontinuous k variate b spline to a k 1 variate b spline with the help of these formulas we have succeeded in determining the exact class of local and global smoothness of a multivariate b spline depending on the configuration of the knots

approximation by cubic c 1 splines on arbitrary triangulations in this paper we present an efficient representation for bivariate piecewise cubicc 1 splines on arbitrary triangulations a numerical method is discussed for computing the dimension of the spaces 3 1 of these splines we consider subspaces ofs 3 1 satisfying certain boundary conditions some applications are given where piecewise cubicc 1 functions are used to solve interpolation problems and least squares approximation problems

on the linear independence of multivariate b splines ii complete configurations the first part of this paper is concerned with global characterizations of both the multivariate b spline and the multivariate truncated power function as smooth piecewise polynomials in the second part of the paper we establish combinatorial criteria for the linear independence of multivariate b splines corresponding to certain configurations of knot sets

bivariate splines on hexagonal lattice for digital image processing in this paper the applications of multi variable spline functions in the digital signal processing was investigated further several two dimensional spline functions were constructed for hexagonal lattice they are actually variation of bivariate cardinal splines on regular triangular partition that were constructed earlier those bivariate cardinal splines have a number of desirable features which make it applicable for image processing and other areas one feature lies in the fact that the coefficients of the interpolation representations with bivariate cardinal splines are basically the sample data at the interpolation points the interpolations converge uniformly to the function being interpolated as the sampling increment approaches zero compare to the popular tensor product with the same degree of smoothness our bivariate cardinal splines has the same degree of accuracy with smaller local support lower degree of the polynomials therefore it should lead to the reduction of computation time and storage space

splines with boundary conditions this paper explores spaces of splines satisfying boundary conditions using the long exact sequence for relative homology we relate them to the boundary complex of d where d is a planar simplicial complex which triangulates a pseudomanifold in this case there is a natural relationship between the hilbert polynomials which measure the dimension of c k r d for k 0 of the spline modules

homogeneous simplex splines homogeneous simplex splines also known as cone splines or multivariate truncated power functions are discussed from a perspective of homogeneous divided differences and polar forms this makes it possible to derive the basic properties of these splines in a simple and economic way in addition a construction of spaces of homogeneous simplex splines is considered which in the nonhomogeneous setting is due to dahmen micchelli and seidel a proof for this construction is presented based on knot insertion restricting the homogeneous splines to a sphere gives rise to spaces of spherical simplex splines

degenerations of ideal hyperbolic triangulations let m be a cusped 3 manifold and let mathcal t be an ideal triangulation of m the deformation variety mathfrak d mathcal t a subset of which parameterises incomplete hyperbolic structures obtained on m using mathcal t is defined and compactified by adding certain projective classes of transversely measured singular codimension one foliations of m this leads to a combinatorial and geometric variant of well known constructions by culler morgan and shalen concerning the character variety of a 3 manifold

subdivision algorithms converge quadratically this paper is concerned with determining the exact convergence rates of subdivision algorithms for various settings of polynomial and spline curve or surface representations

multivariate b splines on triangulated rectangles although many results in univariate spline theory have been extended to the higher dimensional settings by taking tensor products very little is known on the general theory of multivariate spline functions since a univariate spline function is a smooth piecewise polynomial separated by a set of points which are called knots a bivariate spline function is a smooth piecewise polynomial in two variables separated by a grid of curves and so on in the two dimensional setting for example if a domain g in ifi is divided into a finite or countable number of cells by a grid partition a then the space s a of multivariate or more precisely bivariate spline functions is the collection of all functions in p u such that the restriction of every s e s a to each cell of the grid partition is a polynomial p u 4 of total degree k namely

grid point interpolation on finite regions using c 1 box splines multivariate grid point interpolation on finite regions is considered by translates of c 1 box splines defined on a s 1 direction mesh in mathbb r s in general this problem will give more degrees of freedom than the number of interpolation conditions and hence the problem has no unique solution among all interpolants the one minimizing a smoothing functional is chosen two choices of the smoothing functional are proposed and the related existence and uniqueness problems are studied examples showing constructions of box spline surfaces on nonrectangular regions in lr2 and rectangular regions in mathbb r 3 are presented the method can also be used to construct a smooth interpolant to a set of points given at an almost arbitrary set of grid points in mathbb r s

on the optimal approximation rates for criss cross finite element spaces abstract let denote the triangulation of the plane obtained by multi integer translates of the four lines x 0 y 0 x y and x y by l k h we mean the space of all piecewise polynomials of degree k with respect to the scaled triangulation h having continuous partial derivatives of order on r 2 we show that the approximation properties of l k h are completely governed by those of the space spanned by the translates of all so called box splines contained in l k h combining this fact with fourier analysis techniques allows us to determine the optimal controlled approximation rates for the above subspace of box splines where is the largest degree of smoothness for which these spaces are dense as h tends to zero furthermore we study the question of local linear dependence of the translates of the box splines for the above criss cross triangulations

approximation from shift invariant subspaces of l 2 ℝ d a complete characterization is given of closed shift invariant subspaces of l2 rd which provide a specified approximation order when such a space is principal i e generated by a single function then this characterization is in terms of the fourier transform of the generator as a special case we obtain the classical strang fix conditions but without requiring the generating function to decay at infinity the approximation order of a general closed shift invariant space is shown to be already realized by a specifiable principal subspace

developments in bivariate spline interpolation the aim of this survey is to describe developments in the field of interpolation by bivariate splines we summarize results on the dimension and the approximation order of bivariate spline spaces and describe interpolation methods for these spaces moreover numerical examples are given

optimal biorthogonal wavelet decomposition of wire frame meshes using box splines and its application to the hierarchical coding of 3 d surfaces optimal mechanisms are determined for the hierarchical decomposition of wire frame surfaces generated by box splines a family of box splines with compact support suitable for the approximation of wire frames is first defined generated by arbitrary sampling matrices with integer eigenvalues for each such box spline the optimal positioning of the wire frame nodes is determined for each level of the hierarchical wire frame decomposition criterion of optimality is the minimization of the variance of the error difference between the original surface and its representation at each resolution level this is needed so as to ensure that the wire mesh produces at each resolution as close a replica of the original surface as possible several such combinations of box spline generated meshes and the corresponding optimal node lattice sequences are examined in detail with a view to practical application their specific application to the hierarchical coding of three dimensional 3 d wire meshes is experimentally evaluated

translates of exponential box splines and their related spaces exponential box splines eb splines are multivariate compactly supported functions on a regular mesh which are piecewise in a space z spanned by exponential polynomials this space can be defined as the intersection of the kernels of certain partial differential operators with constant coefficients the main part of this paper is devoted to algebraic analysis of the space h of all entire functions spanned by the integer translates of an eb spline this investigation relies on a detailed description of z and its discrete analog s the approach taken here is based on the observation that the structure of z is relatively simple when z is spanned by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process also we find it more efficient to apply directly the relevant differential and difference operators rather than the alternative techniques of fourier analysis thus while generalizing the known theory of polynomial box splines the results here offer a simpler approach and a new insight towards this important special case we also identify and study in detail several types of singularities which occur only for complex eb splines the first is when the fourier transform of the eb spline vanishes at some critical points the second is when z cannot be embedded in y and the third is when h is a proper subspace of z we show among others that each of these three cases is strictly included in its former and they all can be avoided by a refinement of the mesh

on the dimension of bivariate superspline spaces etude de la dimension de certains espaces supersplines des limites superieure et inferieure sont donnees

least supported bases and local linear independence 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

lagrange interpolation by bivariate splines on cross cut partitions the piecewise algebraic curve is a generalization of the classical algebraic curve in this paper we give the nother type theorem of piecewise algebraic curves on cross cut partitions by using interpolation along a piecewise algebraic curve we also present a method to construct lagrange interpolation sets for bivariate spline spaces on cross cut partitions

bivariate c 1 quadratic finite elements and vertex splines following work of heindl and of powell and sabin each triangle of an arbitrary regular triangulation a of a polygonal region ii in r is subdi vided into twelve triangles using the three medians yielding the refinement a of a so that c quadratic finite elements can be constructed in this paper we derive the bezier nets of these elements in terms of the parameters that de scribe function and first partial derivative values at the vertices and values of the normal derivatives at the midpoints of the edges of a consequently bivariate c quadratic generalized vertex splines on a have an explicit formulation here a generalized vertex spline is one which is a piecewise polynomial on the refined grid partition a and has support that contains at most one vertex of the original partition a in its interior the collection of all c quadratic general ized vertex splines on a so constructed is shown to form a basis of s2 a the vector space of all functions on c i2 whose restrictions to each triangular cell of the partition a are quadratic polynomials a subspace with the basis given by appropriately chosen generalized vertex splines with exactly one vertex of a in the interior of their supports that reproduces all quadratic polynomials is identified and hence has approximation order three quasi interpolation for mulas using this subspace are obtained in addition a constructive procedure that yields a locally supported basis of yet another subspace with dimension given by the number of vertices of a that has approximation order three is given

comparison of interacting diffusions and an application to their ergodic theory a general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived as an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent and the components take values in an interval bounded on one side the technique also gives an alternative proof of clustering in the case of compact intervals

incompressibility criteria for spun normal surfaces we give a simple sufficient condition for a spun normal surface in an ideal triangulation to be incompressible namely that it is a vertex surface with non empty boundary which has a quadrilateral in each tetrahedron while this condition is far from being necessary it is powerful enough to give two new results the existence of alternating knots with non integer boundary slopes and a proof of the slope conjecture for a large class of 2 fusion knots

box spline interpolation a computational study abstract we present a computational study of box spline interpolation in two space dimensions we are in particular concerned with the problem of computing the box spline coefficients in the presence of large data sets the issues of convergence and regularity of the interpolants are also considered from a computational point of view

spline interpolations over type 2 triangulations this paper presents a structure of bivariate spline interpolation bases on nonuniform type 2 triangulations the approximation properties of spline interpolations are also considered

multivariate spline spaces in principle all of the properties and applications of multivariate spline functions can be derived from the basic theories in chapter 1 the expressions of the multivariate spline functions in 2 of chapter 1 are inconvenient to be applied directly because the expressions also rely on the solutions of the global conformality conditions so as to determine all the smooth cofactors

approximation by smooth multivariate splines abstract one of the important properties of univariate splines is that in most senses smooth splines approximate just as well as do piecewise polynomials on the same mesh this report shows this to be untrue in the multivariate setting in particular it details the cost in approximating power one may have to pay for the luxury of a smooth piecewise polynomial approximant in an extreme case piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or rho continuous will fail to approximate certain smooth functions at all as the grid goes to zero unless rho is kept below r 3 2 during the analysis of approximation on a certain regular triangular grid a novel kind of bivariate b spline is introduced this b spline in contrast to the established multivariate b spline derived from a simplex can be made to have all its breaklines in a given regular grid this makes it a prime candidate for use in the construction of smooth multivariate piecewise polynomial approximation and its properties will be explored further

vector bundles on complex projective spaces and systems of partial differential equations i on etablit et on etudie une relation entre l espace des solutions d un systeme d equations aux derivees partielles a coefficient constant et la cohomologie d un fibre vectoriel associe faisceau reflexif sur un espace projectif complexe

homology of smooth splines generic triangulations and a conjecture of strang for a a triangulated d dimensional region in rd let sr a denote the vector space of all cr functions f on a that restricted to any simplex in a are given by polynomials of degree at most m we consider the problem of computing the dimension of such spaces we develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds a in the plane getting lower bounds on the dimension of sr a for all r for r 1 we prove a conjecture of strang concerning the generic dimension of the space of cl splines over a triangulated manifold in r2 finally we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region

locking effects in the finite element approximation of elasticity problems we consider the finite element approximation of the 2d elasticity problem when the poisson ratiov is close to 0 5 it is well known that the performance of certain commonly used finite elements deteriorates asv 0 a phenomenon calledlocking we analyze this phenomenon and characterize the strength of the locking androbustness of varioush version schemes using triangular and rectangular elements we prove that thep andh p versions are free of locking with respect to the error in the energy norm a generalization of our theory to the 3d problem is also discussed

an explicit basis for c 1 quartic by various bivariate splines we establish the dimension of the space of c 1 bivariate piecewise quartic polynomials defined on a triangulation of a connected polygonal domain our approach is to construct a minimal determining set and an associated explicit basis for the space for general triangulations the minimal determining set must be defined globally

blossoming begets b spline bases built better by b patches the concept of symmetric recursive algorithm leads to new sdimensional spline spaces we present a general scheme for constructing a collection of multivariate b splines with k i continuous derivatives whose linear span contains all polynomials of degree at most k this scheme is different from the one developed earlier by dahmen and micchelli and independently by h6llig which was based on combinatorial principles and the geometric interpretation of the b spline the new spline space introduced here seems to offer possibilities for economizing the computation for evaluating linear combinations of b splines

hyperbolic structures on 3 manifolds i deformation of acylindrical manifolds this is the first in a series of papers showing that haken manifolds have hyperbolic structures this first was published the second two have existed only in preprint form and later preprints were never completed this eprint is only an approximation to the published version which is the definitive form for part i and is provided for convenience only all references and quotations should be taken from the published version since the theorem numbering is different and not all corrections have been incorporated into the present version parts ii and iii will be made available as eprints shortly

a note on bivariate box splines on a k direction mesh abstract we determine the dimension of the polynomial subspace of the linear space spanned by the translates over lattice points of a bivariate box spline on a k direction mesh

translates of multivarlate splines abstract we study spaces generated by translations of a fixed function over lattice points we provide some new algebraic and approximation properties for these spaces which show their applicability for finite element analysis

algebraic phase unwrapping based on two dimensional spline smoothing over triangles two dimensional 2d phase unwrapping is an estimation problem of a continuous phase function over a 2d domain from its wrapped samples in this paper we propose a novel approach for high resolution 2d phase unwrapping in the first step spline smoothing sps we construct a pair of the smoothest spline functions which minimize the energies of their local changes while interpolating respectively the cosine and the sine of given wrapped samples if these functions have no common zero over the domain the proposed estimate of the continuous phase function can be obtained by algebraic phase unwrapping in the second step algebraic phase unwrapping apu to avoid the occurrence of common zeros in sps due to phase noise in the observed wrapped samples we also propose a denoising step denoising by selective smoothing dss as preprocessing which selectively smooths unreliable wrapped samples by using convex optimization the smoothness of the proposed unwrapped phase function is guaranteed globally over the domain without losing any desired consistency with all reliable wrapped samples numerical experiments for terrain height estimation demonstrate the effectiveness of the proposed 2d phase unwrapping scheme

structure and dimension of multivariate spline space of lower degree on arbitrary triangulation in this paper we discuss the structure of multivariate spline spaces on arbitrary triangulation by using the methods and results of smoothing cofactor and generator basis of modules on the base of analyzing the algebraic and geometric results about singularity of s21 ms we build the structure of triangulation and give some useful geometric conditions such that s 1 space is singular and we obtain an algebraic condition which is necessary and sufficient for the singularity of s 1 spaces as well as their dimension formulae moreover the structure matrix of spline spaces over any given partition is defined which has been used to discuss the structure of s31 and s21 spaces over arbitrary triangulation and to prove the nonsingularity of s31 spaces this partially settles a conjecture on the singularity of spline spaces in wang et al multivariate spline and its applications kluwer press dordrecht 2002 academic press beijing 1994 in chinese meanwhile the dimension formulae of s31 s21 spaces and the dimension formulae of s 1 1 spaces are also given in this paper

certain reflexive sheaves on ⁿ_ and a problem in approximation theory this paper establishes a link between certain local problems in the theory of splines and properties of vector bundles and reflexive sheaves on complex projective spaces

bivariate spline interpolation at grid points 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

on structure of bivariate spline space of lower degree over arbitrary triangulation the structure of bivariate spline space over arbitrary triangulation is complicated because the dimension of a multivariate spline space depends not only on the topological property of the triangulation but also on its geometric property a new vertex coding method to a triangulation is introduced in this paper to further study structure of the spline spaces the upper bound of the dimension of spline spaces over triangulation given by l l schumaker is slightly improved via the new vertex coding method the structure of multivariate spline spaces s 2 1 and s 3 1 over arbitrary triangulation are studied via the method of smoothness cofactor and the structure matrix of multivariate spline ring by luo and wang a kind of sufficient conditions on judging non singularity of the s 2 1 and s 3 1 spaces over arbitrary triangulation is given which only depends on the topological property of the triangulation from the sufficient conditions a triangulation strategy is presented at the end of the paper the strategy ensures that the constructed triangulation is non singular or generic for s 2 1 and s 3 1

approximation order from certain spaces of smooth bivariate splines on a three direction mesh determination de l ordre exact d approximation de certains espaces de fonctions splines regulieres a deux variables

infinite systems with locally interacting components in 1963 glauber analyzed a one dimensional model for magnetism it was the first study of the markovian time evolution of a system with infinitely many interacting components in section 1 it will be discussed in the light of recent progress in this field the remaining sections 2 3 and 4 form an introduction to recent joint work with thomas m liggett they concern new types of systems where each component takes on a real value which fluctuates in a way depending linearly on the values of neighboring components the ergodic theory for such systems with finitely many components is the subject of section 2 the results suggest conjectures for the case of infinitely many components stated in section 3 and proved in the joint paper with t liggett ibid section 4 introduces another class of time evolutions whose ergodic behavior may be analyzed by similar methods

multivariate spline and algebraic geometry abstract the purpose of this survey is to emphasize the special relationship between multivariate spline and algebraic geometry we will not only point out the algebraic geometric method of multivariate spline but also the algebraic geometric background and essence of multivariate spline especially we have made an introduction to the so called piecewise algebraic curve piecewise algebraic variety and some of their properties

a bivariate c1 cubic super spline space on powell sabin triangulation a bivariate c 1 cubic super spline is constructed on powell sabin type 1 split with the additional smoothness at vertices in the original triangulation being c 2 which permits the hermite interpolation up to the second order partial derivatives exactly on all the vertices in the original triangulation the locally supported dual basis and computational details using derivatives around each vertex are given

cardinal and scattered cardinal interpolation by functions having non compact support abstract in this paper we introduce the notion scattered cardinal interpolation and obtain a necessary and sufficient condition for the solvability of this interpolation scheme we also generalize some results of de boor hollig and riemenschneider chui jetter and ward and jetter and riemenschneider on the correctness of lp cardinal interpolation on zd or its submoduls

bivariate quartic spline spaces and quasi interpolation operators in this paper we study two bivariate quartic spline spaces s 4 3 2 d m n 2 and s 4 2 3 d m n 2 and present two classes of quasi interpolation operators in the two spaces respectively some results on the operators are given

multilevel refinable triangular psp splines tri psps a multi level spline technique known as partial shape preserving splines psps li and tian 2011 has recently been developed for the design of piecewise polynomial freeform geometric surfaces where the basis functions of the psps can be directly built from an arbitrary set of polygons that partitions a giving parametric domain this paper addresses a special type of psps the triangular psps tri psps where all spline basis functions are constructed from a set of triangles compared with other triangular spline techniques tri psps have several distinctive features firstly for each given triangle the corresponding spline basis function for any required degree of smoothness can be expressed in closed form and directly written out in full explicitly as piecewise bivariate polynomials secondly tri psps are an additive triangular spline technique where the spline function built from a given triangle can be replaced with a set of refined spline functions built on a set of smaller triangles that partition the initial given triangle in addition tri psps are a multilevel spline technique tri psps surfaces can be designed to have a continuously varying levels of detail achieved simply by specifying a proper value for the smoothing parameter introduced in the spline functions in terms of practical implementation tri psps are a parallel computing friendly spline scheme which can be easily implemented on modern programmable gpus or on high performance computer clusters since each of the basis functions of tri psps can be directly computed independent of each other in parallel

on the dimensions of bivariate spline spaces and the stability of the dimensions 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

multivariate discrete splines and linear diophantine equations in this paper we investigate the algebraic properties of multivariate discrete splines it turns out that multivariate discrete splines are closely related to linear diophantine equations in particular we use a solvability condition for a system of linear diophantine equations to obtain a necessary and sufficient condition for the integer translates of a discrete box spline to be linearly independent in order to understand the local structure of discrete splines we develop a general theory for certain systems of linear partial difference equations using this theory we prove that the integer translates of a discrete box spline are locally linearly independent if and only if they are linearly independent

semi cardinal interpolation and difference equations from cubic b splines to a three direction box spline construction this paper considers the problem of interpolation on a semi plane grid from a space of box splines on the three direction mesh building on a new treatment of univariate semi cardinal interpolation for natural cubic splines the solution is obtained as a lagrange series with suitable localization and polynomial reproduction properties it is proved that the extension of the natural boundary conditions to box spline semi cardinal interpolation attains half of the approximation order of the cardinal case

dissipation and high disorder given a fieldfb x gx2zd of independent standard brownian motions indexed by z d the generator of a suitable markov process on z d g and suciently nice function 0 1 0 1 we consider the influence of the parameter on the behavior of the system dut x gut x dt ut x dbt x t 0 x2 z d u0 x c0 0 x we show that for any

homogeneous polynomial splines we construct functions which are piecewise homogeneous polynomials in the positive octant in three dimensions these give a rich and elegant theory which combines properties of polynomial box splines see 6 and the references therein with the explicit representation of simple exponential box splines 11 while enjoying complete symmetry in the three variables by a linear transformation followed by a projection on suitable planes one obtains piecewise polynomial functions of two variables on a mesh formed by three pencils of lines the vertices of these pencils may be finite or one or two may be infinite i e the corresponding pencils may comprise parallel lines as a limiting case all three vertices become infinite and one recovers polynomial box splines on a three direction mesh

the dimension and basis of spaces of multivariate splines abstract the purpose of this paper is to study the recent development of certain aspects of multivariate spline spaces more specifically we will survey the existence of bivariate spline the dimension and basis of the bivariate spline spaces s k with various partitions furthermore we will also introduce some results on the higher dimensional splines

bivariate box splines and smooth pp functions on a three direction mesh abstract let s denote the space of bivariate piecewise polynomial functions of degree k and smoothness on the regular mesh generated by the three directions 1 0 0 1 1 1 we construct a basis for s in terms of box splines and truncated powers this allows us to determine the polynomials which are locally contained in s and to give upper and lower bounds for the degree of approximation for 2k 2 3 k 2 3 the case of minimal degree k for given smoothness we identify the elements of minimal support in s and give a basis for s loc f s supp f with a convex subset of r 2

locking effects in the finite element approximation of plate models we analyze the robustness of various standard finite element schemes for a hierarchy of plate models and obtain asymptotic convergence estimates that are uniform in terms of the thickness d we identify h version schemes that show locking i e for which the asymptotic convergence rate deteriorates as d 0 and also show that the p version is free of locking in order to isolate locking erects from boundary layer erects which also arise as d 0 our analysis is carried out for the periodic case which is free of boundary layers we analyze in detail the lowest model of the hierarchy the well known reissner mindlin model and show that the locking and robustness of finite element schemes for higher models of the hierarchy are essentially identical to the riessner mindlin case