new approach to study splines by blossoming method and application to the construction of a bivariate c 1 quartic quasi interpolant this work is a contribution in the approximation theory for studying and analyzing piecewise polynomial functions splines which uses the blossoming approach some existing results in the literature are reformulated such as the smoothness conditions between polynomials of a spline by using the affinity property of the blossom some definitions of sub splines are proposed which can be very useful in the study and the construction of splines such as macro elements or quasi interpolants as an application of the proposed results a c 1 quartic spline quasi interpolant with optimal approximation order is defined without using any mask for smoothness or b spline basis numerical results are presented and compared with other methods given in the literature

the supremum of some canonical processes 1 introduction consider a random variable r v h such that eh2 l distributed like h to each t tk k 1 of 2 one can associate a r v xi ek l tkhk this defines a stochastic process xt t e2 by stochastic process we simply mean a collection of r v this process could be called the canonical process associated to h a long range and rather ambitious program would be to understand given h for which subsets t of 2 the process xt tct is bounded let us first discuss the case where h is centered gaussian the special importance of that case stems from the fact that for the question of boundedness discussed here gaussian processes of the type xt tct are the most general gaussian processes this fact is closely related to the rotational invariance of gaussian measures for a stochastic process xt tct where the set t is possibly uncountable we set

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

adiabatic perturbations for compactons under dissipation and numerically induced dissipation compacton propagation under dissipation shows amplitude damping and the generation of tails the numerical simulation of compactons by means of dissipative schemes also show the same behaviors the truncation error terms of a numerical method can be considered as a perturbation of the original partial differential equation and perturbation methods can be applied to its analysis for dissipative schemes or when artificial dissipation is added the adiabatic perturbation method yields evolution equations for the amplitude loss in the numerical solution and the amplitude of the numerically induced tails in this paper such methods are applied to the k 2 2 rosenau hyman equation showing a very good agreement between perturbative and numerical results

on the boundedness of bernoulli processes we present a positive solution to the so called bernoulli conjecture concerning the characterization of sample boundedness of bernoulli processes we also discuss some applications and related open problems

near minimally normed spline quasi interpolants on uniform partitions spline quasi interpolants qis are local approximating operators for functions or discrete data we consider the construction of discrete and integral spline qis on uniform partitions of the real line having small infinity norms we call them near minimally normed qis they are exact on polynomial spaces and minimize a simple upper bound of their infinity norms we give precise results for cubic and quintic qis also the qi error is considered as well as the advantage that these qis present when approximating functions with isolated discontinuities

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

computing the range of values of real functions using b spline form abstract in this paper we present an easy and efficient method for computing the range of a function by using spline quasi interpolation we exploit the close relationship between the spline function and its control polygon and use tight subdivision technique in order to obtain monotonic splines which make the range of the spline easy to compute the proposed method is useful in case of given scattered data generated by some unknown function f or scientific measurements several numerical examples are given for cubic and quintic quasi interpolant approximant to illustrate the efficiency and the performance of our method

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

a repository of equations with cosine sine compactons nonlinear evolution equations with cosine sine compacton solutions are reviewed including the rosenau hyman equation and generalizations of korteweg de vries camassa holm boussinesq benjamin bona mahony klein gordon and other equations each equation is generalized to three dimensions and the conditions for its cosine solitary waves to be either a compacton or a soliton are determined several equations claimed in the literature to be different among them are found to be equivalent

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

majorizing measures without measures we give a reformulation of majorizing measures that does not involve measures but rather special sequences of partitions this formulation is more convenient to perform chaining with different distances

a general spline differential quadrature method based on quasi interpolation the differential quadrature method is a numerical discretization technique for the approximation of derivatives the classical method is polynomial based and there is a natural restriction in the number of grid points involved a general spline based method is proposed to avoid this problem for any degree a lagrangian spline interpolant is defined having a fundamental function with small support a quasi interpolant is used to achieve the optimal approximation order that two stage scheme is detailed for the cubic quartic quintic and sextic cases and compared with another methods that appear in the literature

regularity of gaussian processes on donne la validite de la conjecture de fernique 1974 sur l existence de mesures de majoration pouvant caracteriser la limitation des processus gaussiens

increasing the approximation order of spline quasi interpolants abstract in this paper we show how by a very simple modification of bivariate spline discrete quasi interpolants we can construct a new class of quasi interpolants which have remarkable properties such as high order of regularity and polynomial reproduction more precisely given a spline discrete quasi interpolation operator q d which is exact on the space p m of polynomials of total degree at most m we first propose a general method to determine a new differential quasi interpolation operator q r d which is exact on p m r q r d uses the values of the function to be approximated at the points involved in the linear functional defining q d as well as the partial derivatives up to the order r at the same points from this result we then construct and study a first order differential quasi interpolant based on the c 1 cubic b spline on the equilateral triangulation with a hexagonal support when the derivatives are not available or extremely expensive to compute we approximate them by appropriate finite differences to derive new discrete quasi interpolants q d we estimate with small constants the quasi interpolation errors f q r d f and f q d f in the infinity norm finally numerical examples are used to analyze the performance of the method

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

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

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

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

new exact explicit peakon and smooth periodic wave solutions of the k 3 2 equation abstract based on an independent variable transformation the k 3 2 equation is investigated by the bifurcation method of planar systems and qualitative theory of polynomial differential system in different regions of the parametric space some new exact explicit peakon and smooth periodic wave solutions are obtained the results presented in this paper improve the previous results

optimal bivariate c1 cubic quasi interpolation on a type 2 triangulation in a guessab o nouisser g schmeisser multivariate approximation by a combination of modified taylor polynomials j comput appl math 196 2006 162 179 a general method is proposed to increase the approximation order of approximation operators in this work by using these enhancement techniques we introduce and study new schemes based on a c 1 spline quasi interpolant on a type 2 triangulation they are designed for approximating real valued functions defined on r 2 the proposed method is based on the following idea from a discrete quasi interpolant defined by the quadratic box spline exact on p 2 and by judiciously choosing the first order taylor coefficients we derive a cubic differential quasi interpolant yielding optimal approximation order in addition when the derivatives are not available or are extremely expensive to compute we approximate them by finite difference approximations having the desired accuracy to derive a new class of discrete quasi interpolants as an essential difference to some of the existing methods we only use the given data values and then we do not modify the original triangulation finally we present some numerical tests which confirm the efficiency of the newly quasi interpolant and demonstrate good visual quality in particular we compare it with a differential quasi interpolant done by lai m j lai approximation order from bivariate c 1 cubics on a four directional mesh is full comput aided geom design 11 2 1994 215 223 which is also exact on p 3 but uses third order partial derivatives

a finite difference scheme for the k 2 2 compacton equation the k 2 2 equation introduced by rosenau and hyman is a wave equation that possesses solutions compactons that vanish outside a bounded interval of the spatial axis a finite difference scheme for this equation is suggested that can successfully cope with compacton interactions leading to negative waves we show rigorously that in those interactions a loss of smoothness of the solution necessarily takes place

majorizing measures the generic chaining majorizing measures provide bounds for the supremum of stochastic processes they represent the most general possible form of the chaining argument going back to kolmogorov majorizing measures arose from the theory of gaussian processes but they now have applications far beyond this setting the fundamental question is the construction of these measures this paper focuses on the tools that have been developed for this purpose and in particular the use of geometric ideas applications are given to several natural problems where entropy methods are powerless

quasi interpolation by c1 quartic splines on type 1 triangulations abstract in this paper we construct two new families of c 1 quartic quasi interpolating splines on type 1 triangulations approximating regularly distributed data the splines are directly determined by setting their bernstein bezier coefficients to appropriate combinations of the given data values instead of defining the approximating splines as linear combinations of compactly supported bivariate spanning functions and do not use prescribed derivatives at any point of the domain the quasi interpolation operators provided by the proposed schemes interpolate the data values at the vertices of the triangulation reproduce cubic polynomials and yield approximation order four for smooth functions we also propose some numerical tests that confirm the theoretical results

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

self similar radiation from numerical rosenau hyman compactons the numerical simulation of compactons solitary waves with compact support is characterized by the presence of spurious phenomena as numerically induced radiation which is illustrated here using four numerical methods applied to the rosenau hyman k p p equation both forward and backward radiations are emitted from the compacton presenting a self similar shape which has been illustrated graphically by the proper scaling a grid refinement study shows that the amplitude of the radiations decreases as the grid size does confirming its numerical origin the front velocity and the amplitude of both radiations have been studied as a function of both the compacton and the numerical parameters the amplitude of the radiations decreases exponentially in time being characterized by a nearly constant scaling exponent an ansatz for both the backward and forward radiations corresponding to a self similar function characterized by the scaling exponent is suggested by the present numerical results

a multinomial spline approximation scheme using spline quasi interpolants abstract this paper presents a multinomial spline approximation scheme based on spline quasi interpolants the scheme can be considered as an extension of the usual bernstein approximation for complex exponentials error estimates and numerical examples are given to show that this new scheme could produce highly accurate results

on the construction of trivariate near best quasi interpolants based on c 2 quartic splines on type 6 tetrahedral partitions the construction of new quasi interpolants qis having optimal approximation order and small infinity norm and based on a trivariate c 2 quartic box spline is addressed in this paper these quasi interpolants called near best qis are obtained in order to be exact on the space of cubic polynomials and to minimize an upper bound of their infinity norm which depends on a finite number of free parameters in a tetrahedral sequence defining the coefficients of the qis we show that this problem has always a unique solution which is explicitly given we also prove that the sequence of the resulting near best quasi interpolants converges in the infinity norm to the schoenberg operator

superconvergent local quasi interpolants based on special multivariate quadratic spline space over a refined quadrangulation abstract in this paper we first recall some results concerning the construction and the properties of quadratic b splines over a refinement of a quadrangulation of a planar domain introduced recently by lamnii et al then we introduce the b spline representation of hermite interpolant in the special space s 2 1 0 of any polynomial or any piecewise polynomial over refined quadrangulation of after that we use this b representation for constructing several superconvergent discrete quasi interpolants the new results that we present in this paper are an improvement and a generalization of those developed in the above cited paper

local discontinuous galerkin methods for nonlinear dispersive equations we develop local discontinuous galerkin dg methods for solving nonlinear dispersive partial differential equations that have compactly supported traveling waves solutions the so called compactons the schemes we present extend the previous works of yan and shu on approximating solutions for linear dispersive equations and for certain kdv type equations we present two classes of dg methods for approximating solutions of such pdes first we generate nonlinearly stable numerical schemes with a stability condition that is induced from a conservation law of the pde an alternative approach is based on constructing linearly stable schemes i e schemes that are linearly stable to small perturbations the numerical simulations we present verify the desired properties of the methods including their expected order of accuracy in particular we demonstrate the potential advantages of using dg methods over pseudospectral methods in situations where discontinuous fronts and rapid oscillations co exist in a solution

a theorem on majorizing measures let t d be a metric space and r 1r an increasing convex function with 0 0 we prove that if m is a probability measure m on r which is majorizing with respect to d that is supper d t 0 1 1 m b x e de then e sup x s x t 32 s t t for each separable stochastic process x t t t which satisfies e x s x t d s t 1 for all s t t s t this is a strengthening of one of the main results from talagrand ann probab 18 1990 1 491 and its proof is significantly simpler

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

stability and dynamical properties of rosenau hyman compactons using pade approximants we present a systematic approach for calculating higher order derivatives of smooth functions on a uniform grid using pade approximants we illustrate our findings by deriving higher order approximations using traditional second order finite difference formulas as our starting point we employ these schemes to study the stability and dynamical properties of k 2 2 rosenau hyman compactons including the collision of two compactons and resultant shock formation our approach uses a differencing scheme involving only nearest and next to nearest neighbors on a uniform spatial grid the partial differential equation for the compactons involves first second and third partial derivatives in the spatial coordinate and we concentrate on four different fourth order methods which differ in the possibility of increasing the degree of accuracy or not of one of the spatial derivatives to sixth order a method designed to reduce round off errors was found to be the most accurate approximation in stability studies of single solitary waves even though all derivates are accurate only to fourth order simulating compacton scattering requires the addition of fourth derivatives related to artificial viscosity for those problems the different choices lead to different amounts of spurious radiation and we compare the virtues of the different choices

construction of spherical spline quasi interpolants based on blossoming a general theory of quasi interpolants based on quadratic spherical powell sabin splines on spherical triangulations of a sphere like surface s is developed by using polar forms as application various families of discrete and differential quasi interpolants reproducing quadratic spherical bezier bernstein polynomials or the whole space of the spherical powell sabin quadratic splines of class c 1 are presented

numerical methods based on modified equations for nonlinear evolution equations with compactons abstract compactons are traveling wave solutions with compact support resulting from the balance of both nonlinearity and nonlinear dispersion numerical methods with second fourth sixth and eighth order approximations to the spatial derivatives obtained by means of the method of modified equations applied to the ismail taha finite difference scheme for the rosenau hyman equation are developed the whole set of methods is compared among them in accuracy invariant conservation and in compacton collisions the best method among those studied in terms of the tradeoff between accuracy and computational cost is determined

time stepping in petrov galerkin methods based on cubic b splines for compactons abstract four numerical methods with first to fourth order of accuracy have been developed for the time integration of the rosenau hyman k 2 2 equation the error in the solution and the invariants for the propagation of one compacton and the stability in collisions among compactons have been studied using these methods numerically induced radiation has also been characterized by means of wavefront velocity and wavefront amplitude showing that the self similarity of the radiation wavepackets observed in the numerical results is a consequence of the time stepping method among the four methods studied in this paper the best results in terms of accuracy computational cost and stability have been obtained by means of using the second order time integration method

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