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

a general procedure for the adaptation of multistep algorithms to the integration of oscillatory problems this paper introduces a general technique for the construction of multistep methods capable of integrating without local truncation error homogeneous linear odes with constant coefficients including those in particular that result in oscillatory solutions moreover these methods can be further adapted through coefficient modification for the exact integration of forced oscillations in one or more frequencies even confluent ones that occur from nonhomogeneous terms in the differential equation our procedure allows the derivation of many of the existing codes with similar properties as well as the improvement of others that in their original design were only able to integrate oscillations in a single frequency the properties of the methods are studied within a general framework and numerical examples are presented these demonstrate the way in which the new algorithms perform distinctly better than the general purpose codes particularly when integrating the class of equations with perturbed oscillatory solutions the methods developed are mainly applicable to the accurate and efficient integration of problems for which the oscillation frequencies are known as occurs in satellite orbit propagation the underlying ideas have already been applied to the improvement of some chebyshev methods that are not multistep

algorithms for cardinal interpolation using box splines and radial basis functions we describe an algorithm for bivariate cardinal interpolation which can be applied to translates of basis functions which include box splines or radial basis functions the algorithm is based on a representation of the fourier transform of the fundamental interpolant hence fast fourier transform methods are available in numerical tests the 4 directional box spline transformed to the characteristical submodule of 2 the thin plate spline and the multiquadric case give comparably equal and good results

structure generating mechanisms in agent based models abstract the emergence of dynamical structures in multi agent systems is analysed three different mechanisms are identified namely 1 sensitive dependence in the agent dynamics and convex coupling 2 sensitive dependence and extremal dynamics and 3 interaction through a collectively generated field the dynamical origin of the emergent structures is traced back either to a modification by interaction of the lyapunov spectrum or to multistable dynamics

extrapolation vs projection methods for linear systems of equations it is shown that the four vector extrapolation methods minimal polynomial extrapolation reduced rank extrapolation modified minimal polynomial extrapolation and topological epsilon algorithm when applied to linearly generated vector sequences are krylov subspace methods and are equivalent to some well known conjugate gradient type methods a unified recursive method that includes the conjugate gradient conjugate residual and generalized conjugate gradient methods is developed finally the error analyses for these methods are unified and some known and some new error bounds for them are given

hex splines a novel spline family for hexagonal lattices this paper proposes a new family of bivariate nonseparable splines called hex splines especially designed for hexagonal lattices the starting point of the construction is the indicator function of the voronoi cell which is used to define in a natural way the first order hex spline higher order hex splines are obtained by successive convolutions a mathematical analysis of this new bivariate spline family is presented in particular we derive a closed form for a hex spline of arbitrary order we also discuss important properties such as their fourier transform and the fact they form a riesz basis we also highlight the approximation order for conventional rectangular lattices hex splines revert to classical separable tensor product b splines finally some prototypical applications and experimental results demonstrate the usefulness of hex splines for handling hexagonally sampled data

spectral properties of the transition operator associated to a multivariate refinement equation given a finitely supported sequence a on z s and an s s dilation matrix m the transition operator ta is the linear operator defined by tavoau a p b2z s aomabuvobu where a2 z s and v lies in 0oz s u the linear space of all finitely supported sequences on z s in this paper we investigate the spectral properties of the transition operator ta and apply these properties to the study of the approximation and smoothness properties of the normalized solution of the refinement equation a p a2z s aoau omau 1999

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

saturation theorems for interpolation and the bernstein schnabl operator we shall study properties of box spline operators cardinal interpolation convolution and the bernstein schnabl operator we prove the saturation theorem

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

fundamental solutions for multivariate difference equations soit b une fonction a support compact definie sur le reseau entier z a on considere l equation aux diferences b a k j zd b k j a j f k k z d

simple deterministic self organized critical system we introduce a continuous cellular automaton that presents self organized criticality it is one dimensional totally deterministic without any embedded randomness not even in the initial conditions this system is in the same universality class as the oslo rice pile boundary driven interface depinning and the train model for earthquakes although the system is chaotic in the thermodynamic limit chaos occurs only in a microscopic level

on the a stability of runge kutta collocation methods based on orthogonal polynomials some results about the a stability of runge kutta collocation methods based on the zeros and extrema of the ultraspherical or gegenbauer polynomials are presented these methods are implicit and the a stability is an important property usually demanded the routh hurwitz algorithm is used to establish the main results about the a stability of the methods depending on the parameter lambda of the polynomials for low degree methods some analytical results are presented while for higher degree an open problem based on numerical tests is proposed

discrete truncated powers and lattice points in rational polytope discrete truncate power is very useful for studying the number of nonnegative integer solutions of linear diophantine equations in this paper some detail information about discrete truncated power is presented to study the number of integer solutions of linear diophantine inequations the generalized truncated power and generalized discrete truncated power are defined and discussed respectively we use generalized discrete truncated powers and multivariate splines to investigate the lattice points in rational polytopes in particular we present the degree and period of multidimensional ehrhart polynomial

a general method for constructing quasi interpolants from b splines a general method for constructing quasi interpolation operators based on b splines is developed given a b spline f in r s s 1 normalized by i z email protected i 1 the classical structure q f email protected i z email protected i f i for a quasi interpolation operator q is considered a minimization problem is derived from an estimate of the quasi interpolation error associated with q when lf is a linear combination of values of f at points in some neighbourhood of the support of f or a linear combination of values of f and some of its derivatives at some points in this set or a linear combination of weighted mean values of the function to be approximated that linear functional is defined to produce a quasi interpolant exact on the space of polynomials of maximal total degree included in the space spanned by the integer translates of f the solution of that minimization problem is characterized in terms of specific splines which do not depend on l but only on f

power spectrum of mass and activity fluctuations in a sandpile we consider a directed abelian sandpile on a strip of size 2 ifmmode times else texttimes fi n driven by adding a grain randomly at the left boundary after every t timesteps we establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeros in the ternary base representation of the position of a random walker on a ring of size 3 n we find that while the fluctuations of mass have a power spectrum that varies as 1 f for frequencies in the range 3 ensuremath 2n ensuremath ll f ensuremath ll 1 t the activity fluctuations in the same frequency range have a power spectrum that is linear in f

moving least square reproducing kernel method part ii fourier analysis in part i of this work the moving least square reproducing kernel mlsrk method is formulated and implemented based on its generic construction an m consistency structure is discovered and the convergence theorems are established in this part of the work a systematic fourier analysis is employed to evaluate and further establish the method the preliminary fourier analysis reveals that the mlsrk method is stable for sufficiently dense non degenerated particle distribution in the sense that the kernel function family satisfies the riesz bound one of the novelties of the current approach is to treat the mlsrk method as a variant of the standard finite element method and depart from there to make a connection with the multiresolution approximation in the spirits of multiresolution analysis we propose the following mlsrk transformation fm k hu i 1npu k ik h x xi x w1 the highlight of this paper is to embrace the mlsrk formulation with the notion of the controlled fp approximation based on its characterization the strang fix condition for example a systematic procedure is proposed to design new window functions so they can enhance the computational performance of the mlsrk algorithm the main effort here is to obtain a constant correction function in the interior region of a general domain i e c h 1 this can create a leap in the approximation order of the mlsrk algorithm significantly if a highly smooth window function is embedded within the kernel one consequence of this development is the synchronized convergence phenomenon a unique convergence mechanism for the mlsrk method i e by properly tuning the dilation parameter the convergence rate of higher order error norms will approach the same order convergence rate of the l2 error norm they are synchronized

a class of collocation methods for numerical integration of initial value problems for the numerical solution of initial value problems a general procedure to determine global integration methods is derived and studied they are collocation methods which can be easily implemented and provide a high order accuracy they further provide globally continuous differentiable solutions computation of the integrals which appear in the coefficients are generated by a recurrence formula and no integrals are involved in the calculation numerical experiments provide favorable comparisons with other existing methods

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

interpolation by multivariate splines a general interpolation scheme by multivariate splines at regular sample points is introduced this scheme guarantees the local optimal order of approximation to sufficiently smooth data functions a discussion on numerical implementation is included 1 introduction in this paper we introduce a very general interpolation scheme by multivariate splines based on the quasi interpolation formulas devel oped in 3 we show that the interpolating multivariate splines so obtained give the optimal order of approximation to sufficiently smooth functions let q be a nonnegative locally supported piecewise polynomial function sym metric with respect to the origin and s the linear span of all the translates q j j e z8 of q hence s is a multivariate spline space on a certain grid partition a with certain smoothness joining conditions and of certain total degree induced by q denote by q the fourier transform of q we assume that q is normalized that is

minimizing the quasi interpolation error for bivariate discrete quasi interpolants we define a class of discrete quasi interpolants based on bivariate box splines by imposing the exactness on a space of polynomials of total degree depending on the box spline and minimizing a constant appearing in the leading term of an appropriate quasi interpolation error estimate we give some c 1 quadratic and c 2 quartic examples and compare them with other well known quasi interpolants

variable stepsize chebyshev type methods for the integration of second order i v p s panovsky and richardson a family of implicit chebyshev methods for the numerical integration of second order differential equations j comput appl math 23 1988 35 51 presented a method based on chebyshev approximations for numerically solving the problem y f x y being the steplength constant coleman and booth analysis of a family of chebyshev methods for y f x y j comput appl math 44 1992 95 114 made an analysis of the above method and suggested the convenience to design a variable steplength implementation as far as we know this goal has not been achieved until now later on we extended the above method this journal 2003 and obtained a scheme for numerically solving the equation y 2gy g 2 w 2 f x y the question of how to extend these formulas to variable stepsize procedures is the primary topic of this paper

chebyshev collocation methods for fast orbit determination we apply some chebyshev collocation methods for fast and accurate determination of ephemeris of artificial satellites this formulation is based on the approximation of the differential system by means of chebyshev polynomials in order to assess the convergence of the iterative process and to reduce the number of operations the special formulation of dziobek brouwer is used from several numerical tests we conclude that the method presents very good behaviour for geostationary and low eccentric orbits the most frequent cases the number of operations with this method in some cases is 20 times less than that with a standard integrator besides a typical dense output in a polynomial form is obtained

effortless quasi interpolation in hierarchical spaces we present a general and simple procedure to construct quasi interpolants in hierarchical spaces such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement the proposed hierarchical quasi interpolants are described in terms of the so called truncated hierarchical basis assuming a quasi interpolant is selected for each space associated with a particular level in the hierarchy the hierarchical quasi interpolants are obtained without any additional manipulation the main properties like polynomial reproduction of the quasi interpolants selected at each level are locally preserved in the hierarchical construction we show how to construct hierarchical local projectors and the local approximation order of the underling hierarchical space is also investigated the presentation is detailed for the truncated hierarchical b spline basis and we discuss its extension to a more general framework

dissipative chebyshev exponential fitted methods for numerical solution of second order differential equations a family of implicit methods based on intra step chebyshev interpolation has been developed to integrate oscillatory second order initial value problems of the form y t 2g y t g2 w2 y t f t y t the procedure integrates the homogeneous part exactly in the absence of round off errors the chebyshev approach uses stepsizes that are considerably larger than those typically used in runge kutta or multistep methods computational overheads are comparable to those incurred by high order conventional procedures chebyshev interpolation coupled with the exponential fitted nature of the method substantially reduces local errors global error propagation rates are also reduced making these procedures good candidates to be used in long term simulations of perturbed oscillatory systems with a dissipative term

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

bivariate interpolation with quadratic box splines etablissement des resultats d existence et d unicite pour l interpolation a deux variables par des fonctions splines quadratiques encapsulees

on multivariate e splines abstract in a recent interesting paper a ron 14 initiated the study of cardinal exponential splines in several variables beyond settling some open problems which were suggested by his results this paper addresses a variety of additional questions of an algebraic nature thus some of the results we obtained elsewhere on cube spline surfaces appear now as a special case of the present general setting our approach is critically based on some new bounds and formulas for the dimension of certain intersections of null spaces of commuting linear operators where the particular way of forming these intersections is determined by a matroid structure specifically this allows us to characterize the local linear independence of translates of the exponential cube spline as well as to study local algebraic properties of functions spanned by them only special cases were covered by ron s investigation furthermore we consider in detail the construction of dual bases linear projectors linear dependence relations among these exponential splines and subdivision algorithms for their approximate evaluation the analysis of these latter algorithms is based on the concept of discrete exponential splines we investigate to what extent these discrete objects possess analogous properties to their continuous counterparts finally the combinatorial interpretation of the discrete splines their relationship to their continuous counterparts and the machinery we develop leads to interesting combinatorial applications for enumerating the number of solutions to linear diophantine systems and corresponding reciprocity relations an announcement of these results appeared in 10

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

optimal interpolatory subdivision schemes in multidimensional spaces we analyze the approximation and smoothness properties of fundamental and refinable functions that arise from interpolatory subdivision schemes in multidimensional spaces in particular we provide a general construction of bivariate interpolatory refinement masks such that the corresponding fundamental and refinable functions attain the optimal approximation order and smoothness order in addition these interpolatory refinement masks are minimally supported and enjoy full symmetry several examples are explicitly computed

quasi interpolants associated with h splines on a three direction mesh h splines are b splines with regular hexagonal supports on a three direction mesh of the plane this family contains in particular classical box splines which are convolution powers of the piecewise affine pyramid in this paper we construct discrete or integral qua si interpolants having the best possible approximation order for the smooth functions the construction is based on properties of hexagonal sequences and associated difference operators and of discrete laplacian schemes

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

the number of solutions to linear diophantine equations and multivariate splines etude de la variation du nombre des solutions entieres non negatives de s equations diophantiennes lineaires en fonction des termes non homogenes

shape preserving quasi interpolation and interpolation by box spline surfaces this paper is devoted to the study of shape preserving approximation and interpolation of functions by box spline surfaces on three and four directional meshes the properties of positivity monotonicity and convexity are considered a characterization of the grid spacing is given which guarantees the preservation of these properties for functions in certain lipschitz classes

a family of implicit chebyshev methods for the numerical integration of second order differential equations abstract a family of implicit methods based on intra step chebyshev interpolation is developed for the solution of initial value problems whose differential equations are of the special second order form y f y x x the general procedure allows stepsizes which are considerably larger than commonly used in conventional methods computation overhead is comparable to that required by high order single or multistep procedures in addition the iterative nature of the method substantially reduces local errors while maintaining a low rate of global error growth

cardinal interpolation by multivariate splines the purpose of this paper is to investigate cardinal interpolation using locally supported piecewise polynomials in particular the notion of a commutator is introduced and its connection with the marsden identity is observed the order of a commutator is shown to be equivalent to the strang and fix conditions that arise in the study of the local approxima tion orders using quasi interpolants we also prove that scaled cardinal interpolants give these local approximation orders introduction cardinal interpolation by bivariate box splines was first studied by de boor hollig and riemenschneider 3 the purpose of our paper is to investigate the cardinal interpolation problem from a different point of view in particular the notion of a commutator is introduced it will be shown that this notion generalizes the marsden identity for univariate splines to the multivariate setting the order of a commutator will be shown to be equivalent to the strang and fix conditions used in the study of the order of controlled approximation by dahmen and micchelli 8 or local approximation by de boor and jia 4 an application to obtain approximation orders through the constructive method of scaled cardinal interpolation will also be studied in this paper 1 preliminaries this section consists of preliminary material for multivariate cardinal interpolation our approach is motivated by the work in 5 and 13 where cardinal interpolation in l2 was connected with certain convolution operators lr let z denote the set of integers and z the nonnegative ones for any given complex sequence by

properties and implementation of ρ adams methods based on mixed type interpolation we investigate the properties of the coefficients of modified r adams methods for the integration of odes the derivation of these methods is in contrast with the classical adams methods not based on a polynomial interpolation theory but rather starts from a mixed interpolation theory in which a parameter a is involved it will be shown that the coefficients of the modified methods possess properties which make these methods very attractive further we will discuss the role of so called over implicit modified r adams schemes in the construction of more general linear multistep methods our second goal is to show that the modified adams bashforth adams moulton methods axe very well suited to be implemented as a predictor corrector pair in particular we will discuss the choice of the interpolation parameter when such a method is applied to general systems of equations numerical tests are performed to support the theory

families of methods for ordinary differential equations based on trigonometric polynomials abstract we consider the construction of methods based on trigonometric polynomials for the initial value problems whose solutions are known to be periodic it is assumed that the frequency w can be estimated in advance the resulting methods depend on a parameter hw where h is the step size and reduce to classical multistep methods if 0 gautschi 4 developed adams and stormer type methods in our paper we construct nystrom s and milne simpson s type methods numerical experiments show that these methods are not sensitive to changes in w but require the jacobian matrix to have purely imaginary eigenvalues

characterization of smoothness of multivariate refinable functions in sobolev spaces wavelets are generated from refinable functions by using multiresolution anlalysis in this paper we investigate the srloothness properties of multivariate refinable functionis in sobolev spaces we characterize the optinlal smoothness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite dimensional invariant subspace several examples are provided to illustrate the general theory

zero dissipative semi implicit hybrid method for solving oscillatory or periodic problems in this paper a new semi implicit two step hybrid method with fifth algebraic order is derived for the integration of second order oscillatory initial value problems the new method possesses dispersion of order eight and dissipation of order infinity numerical experiment reveals the superiority of the new method for solving oscillatory or periodic problems over several methods of the same algebraic order in the literature

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

using the refinement equation for evaluating integrals of wavelets the wavelet galerkin method for solving partial differential equations leads to the problem of computing integrals of products of derivatives of wavelets this paper studies the problem from the point of view of stationary subdivision schemes one of the main results is to identify these integrals as components of the unique solution of a certain eigenvector moment problem associated with the coefficients of the refinement equation asymptotic expansions for the corresponding subdivision schemes form an important ingredient of our approach

near best quasi interpolants associated with h splines on a three direction mesh spline quasi interpolants with optimal approximation orders and small norms are useful in several applications in this paper we construct the so called near best discrete and integral quasi interpolants based on h splines i e b splines with regular hexagonal supports on the uniform three directional mesh of the plane these quasi interpolants are obtained so as to be exact on some space of polynomials and to minimize an upper bound of their infinite norms which depend on a finite number of free parameters we show that this problem has always a solution which is not unique in general concrete examples of these types of quasi interpolants are given in the two last sections

simplicial grid refinement on freudenthal s algorithm and the optimal number of congruence classes in the present paper we investigate freudenthal s simplex refinement algorithm which can be considered to be the canonical generalization of bank s well known red refinement strategy for triangles freudenthal s algorithm subdivides any given n simplex into 2 n subsimplices in such a way that recursive application results in a stable hierarchy of consistent triangulations our investigations concentrate in particular on the number of congruence classes generated by recursive refinements after presentation of the method and the basic ideas behind it we will show that freudenthal s algorithm produces at most n 2 congruence classes for any initial n simplex no matter how many subsequent refinements are performed moreover we will show that this number is optimal in the sense that recursive application of any affine invariant refinement strategy with 2 n sons per element results in at least n 2 congruence classes for almost all n simplices

analysis of a family of chebyshev methods for y f x y abstract methods proposed by panovsky and richardson this journal 1988 are interpreted as a family of symmetric two step hybrid methods each method is based on a polynomial interpolant of degree n for y for which the nodes are determined by the extrema of the chebyshev polynomial of degree n it is shown that the order of accuracy is n 1 for odd n and n 2 for even n and expressions for the local error constants are given the stability properties of each method are determined by the roots of a quadratic equation 2 2 n 1 0 where n 2 is a rational approximation for cos the intervals of periodicity which are also intervals of absolute stability are tabulated for n 10 numerical examples provide comparisons with other methods

cardinal hermite interpolation with box splines ii in the present work we extent the results in rs on chip i e cardinal hermite interpolation by the span of translates of directional derivatives of a box spline these directional derivatives are that ones which define the type of the hermite interpolation we admit here several linearly independent directions with multiplicities instead of one direction as in rs under the same assumptions on the smoothness of the box spline and its defining matrixt we can prove as in rs chip has a system of fundamental solutions which are inl l 2 together with its directional derivatives mentioned above moreover for data sequences inl p d 1 p 2 there is a spline function inl p 1 p 1 p 1 which solves chip

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

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

a remark on the convolution with the box spline the semi discrete convolution with the box spline is an important tool in approximation theory we give a formula for the difference between semidiscrete convolution and convolution with the box spline this formula involves multiple bernoulli polynomials 1 box splines and semi discrete convolution let v be a n dimensional real vector space equipped with a lattice if we choose a basis of the lattice then we may identify v with rn and with zn we choose here the lebesgue measure dv associated to the lattice let x a1 a2 an be a sequence a multiset of n nonzero vectors in the zonotope z x associated with x is the polytope

the toeplitz theorem and its applications to approximation theory and linear pdes we take an algebraic approach to the problem of approximation by dilated shifts of basis functions given a finite collection d of compactly supported functions in lp ris 1 0 where sh is the hdilate of s we prove that sh h 0 provides lp approximation order r only if s contains all the polynomials of total degree less than r in particular in the case where d consists of a single function p with its moment f p 0 we characterize the approximation order of sh h 0 by showing that the above condition on polynomial containment is also sufficient the above results on approximation order are obtained through a careful analysis of the structure of shift invariant spaces it is demonstrated that a shiftinvariant space can be described by a certain system of linear partial difference equations with constant coefficients such a system then can be reduced to an infinite system of linear equations whose solvability is characterized by an old theorem of toeplitz thus the toeplitz theorem sheds light into approximation theory it is also used to give a very simple proof for the well known ehrenpreis principle about the solvability of a system of linear partial differential equations with constant coefficients

optimal lattices for sampling the generalization of the sampling theorem to multidimensional signals is considered with or without bandwidth constraints the signal is modeled as a stationary random process and sampled on a lattice exact expressions for the mean square error of the best linear interpolator are given in the frequency domain moreover asymptotic expansions are derived for the average mean square error when the sampling rate tends to zero and infinity respectively this makes it possible to determine the optimal lattices for sampling in the low rate sampling case or equivalently for rough processes the optimal lattice is the one which solves the packing problem whereas in the high rate sampling case or equivalently for smooth processes the optimal lattice is the one which solves the dual packing problem in addition the best linear interpolation is compared with ideal low pass filtering cardinal interpolation

semi infinite multi index perturbed block toeplitz systems in this article banach algebra techniques are employed to study the numerical solution of linear systems with a semi infinite multi index suitably perturbed k k block toeplitz matrix decay properties of their solutions are studied by using suitably weighted wiener algebras a projection type method for their numerical solution is introduced numerical results are presented illustrating both the accuracy of the method for certain perturbed toeplitz systems and the need for generalizing the theoretical framework to other such systems 2003 elsevier science inc all rights reserved

voronoi splines we introduce a framework for construction of non separable multivariate splines that are geometrically tailored for general sampling lattices voronoi splines are b spline like elements that inherit the geometry of a sampling lattice from its voronoi cell and generate a lattice shift invariant spline space for approximation in rd the spline spaces associated with voronoi splines have guaranteed approximation order and degree of continuity by exploiting the geometric properties of voronoi polytopes and zonotopes we establish the relationship between voronoi splines and box splines which are used for a closed form characterization of the former for cartesian lattices voronoi splines coincide with tensor product b splines and for the 2 d hexagonal lattice the proposed approach offers a reformulation of hex splines in terms of multi box splines while the construction is for general multidimensional lattices we particularly characterize bivariate and trivariate voronoi splines for all 2 d and 3 d lattices and specifically study them for body centered cubic and face centered cubic lattices

on near best discrete quasi interpolation on a four directional mesh spline quasi interpolants are practical and effective approximation operators in this paper we construct qis with optimal approximation orders and small infinity norms called near best discrete quasi interpolants which are based on w splines i e b splines with octagonal supports on the uniform four directional mesh of the plane these quasi interpolants are exact on some space of polynomials and they minimize an upper bound of their infinity norms depending on a finite number of free parameters we show that this problem has always a solution in general nonunique concrete examples of such quasi interpolants are given in the last section

the chebyshev methods of panovsky and richardson as runge kutta nystro m methods abstract the two step methods proposed by panovsky and richardson this journal 1988 and analysed by coleman and booth this journal 1992 are shown to be equivalent to certain runge kutta nystrom methods for differential equations of the form y f x y an important advantage of the new formulation is that it is no longer necessary to provide additional starting values at the beginning of a calculation or when the steplength or the order is changed the new formulation is also more stable with respect to the propagation of rounding errors

symmetric magic squares and multivariate splines abstract we use multivariate splines to investigate linear diophantine equations and related problems in graph theory in particular we solve a conjecture of stanley about symmetric magic squares

a characterization of multivariate quasi interpolation formulas and its applications let be a compactly supported function on s ands the linear space withgenerator that is s is the linear span of the multiinteger translates of it is well known that corresponding to a generator there are infinitely many quasi interpolation formulas a characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications and in addition provides a clear formulation of the difficult problem of minimally supported quasi interpolants we introduce a generalization of interpolation called interpolation and a notion of higher order quasi interpolation called approximation a characterization of approximants similar to that of quasi interpolants is obtained with similar applications among these applications are estimating least squares approximants without matrix inversion surface fitting to incomplete or semi scattered discrete data and constructing generators with one point quasi interpolation formulas it will be seen that the exact values of the generator at the multi integers s facilitates the above study an algorithm to yield this information for box splines is discussed

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

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

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

a local lagrange interpolation method based on c 1 cubic splines on freudenthal partitions a trivariate lagrange interpolation method based on c 1 cubic splines is described the splines are defined over a special refinement of the freudenthal partition of a cube partition the interpolating splines are uniquely determined by data values but no derivatives are needed the interpolation method is local and stable provides optimal order approximation and has linear complexity

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

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

average sampling in shift invariant subspaces with symmetric averaging functions in this paper we study the reconstruction of functions in shift invariant subspaces from local averages with symmetric averaging functions we present an average sampling theorem for shift invariant subspaces and give quantitative results on the aliasing error and the truncation error we show that every square integrable function can be approximated by its average sampling series as special cases we also obtain new error bounds for regular sampling examples are given

on the solution of certain systems of partial difference equations and linear dependence of translates of box splines on resout des systemes d equations aux differences partielles qui apparaissent dans l etude de la dependance lineaire des translatees de splines en boite

a faber series approach to cardinal interpolation for a compactly supported function 9 in rd we study quasiinterpolants based on point evaluations at the integer lattice we restrict ourselves to the case where the coefficient sequence af for given data f is computed by applying a univariate polynomial q to the sequence i zd and then convolving with the data flzd such operators appear in the well known neumann series formulation of quasi interpolation a criterion for the polynomial q is given such that the corresponding operator defines a quasi interpolant since our main application is cardinal interpolation which is well defined if the symbol of p does not vanish we choose q as the partial sum of a certain faber series this series can be computed recursively by this approach we avoid the restriction that the range of the symbol of p must be contained in a disk of the complex plane excluding the origin which is necessary for convergence of the neumann series furthermore for symmetric a we prove that the rate of convergence to the cardinal interpolant is superior to the one obtainable from the neumann series

modification of the richardson panovsky methods for precise integration of satellite orbits abstract a family of implicit methods based on intrastep chebyshev interpolation has been developed to integrate initial value problems of the second order harmonic oscillator form d 2 y dt 2 y f y t t the procedure integrates the homogeneous part exactly in the absence of roundoff errors the chebyshev approach uses stepsizes that are considerably larger than those typically used in runge kutta or multistep methods computational overhead is comparable to that incurred by high order conventional procedures chebyshev interpolation coupled with the iterative nature of the method substantially reduces local errors global error propagation rates are also reduced making these procedures good candidates for use in long term simulations of perturbed oscillator systems the procedure is applied to integrations of the ks transformed equations for the oblateness perturbed orbital motion of an artificial satellite

exponentially weighted legendre gauss tau methods for linear second order differential equations abstract we develop a method to solve a class of second order ordinary differential equations with highly oscillatory solutions the method consists in combining three different techniques legendre gauss spectral tau method exponential fitting and coefficient perturbation methods with our approach the resulting approximate solutions are expressed in terms of an exponentially weighted legendre polynomial basis e n x l n x n 0 where n are appropriately chosen complex numbers the accuracy and efficiency of the method are discussed and illustrated numerically

self organized criticality in a bulk driven one dimensional deterministic system abstract we introduce a deterministic self organized critical system that is one dimensional and bulk driven we find that there is no unique universality class associated with the system that is the critical exponents change as the parameters of the system are changed this is in contrast with the boundary driven version of the model phys rev e 61 2000 6056 in which the exponents are unique this model can be seen as a discretized version of the conservative limit of the burridge knopoff model for earthquakes

quasi interpolation projectors for box splines we consider box spline quasi interpolants based on local linear functionals of point evaluator and integral type the approximations are easy to compute and reproduce the whole spline space in question

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

approximation properties of multivariate wavelets wavelets are generated from refinable functions by using multiresolution analysis in this paper we investigate the approximation properties of multivariate refinable functions we give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask we connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators finally we demonstrate that a refinable function in w k 1 1 r s provides approximation order k