compactness characterization of operators in the toeplitz algebra of the fock space fαp abstract for 1 p let t p be the norm closure of the algebra generated by toeplitz operators with bounded symbols acting on the standard weighted fock space f p in this paper we will show that an operator a is compact on f p if and only if a t p and the berezin transform b a of a vanishes at infinity

combining the radial basis function eulerian and lagrangian schemes with geostatistics for modeling of radionuclide migration through the geosphere to assess the long term safety of a radioactive waste disposal system mathematical models are used to describe groundwater flow chemistry and potential radionuclide migration through geological formations a number of processes need to be considered when predicting the movement of radionuclides through the geosphere the most important input data are obtained from field measurements which are not available for all regions of interest for example the hydraulic conductivity as an input parameter varies from place to place in such cases geostatistical science offers a variety of spatial estimation procedures methods for solving the solute transport equation can also be classified as eulerian lagrangian and mixed the numerical solution of partial differential equations pde has usually been obtained by finite difference methods fdm finite element methods fem or finite volume methods fvm kansa introduced the concept of solving partial differential equations using radial basis functions rbf for hyperbolic parabolic and elliptic pdes the aim of this study was to present a relatively new approach to the modeling of radionuclide migration through the geosphere using radial basis function methods in eulerian and lagrangian coordinates in this study we determine the average and standard deviation of radionuclide concentration with regard to variable hydraulic conductivity which was modelled by a geostatistical approach radionuclide concentrations will also be calculated in heterogeneous and partly heterogeneous 2d porous media

estimation of aquifer parameters under transient and steady state conditions 2 uniqueness stability and solution algorithms paper 2 of this three part series starts with a discussion of the question under what conditions is the aquifer inverse problem well posed after defining the terms uniqueness identifiability and stability theoretical considerations and synthetic examples are used to demonstrate that ill posedness can be mitigated by including prior information about the parameters in the estimation criterion to be minimized at the same time the inclusion of such information is shown to be insufficient to guarantee uniqueness and stability in all cases several test problems in the recent literature which have resulted in pessimistic conclusions about the solvability of the aquifer inverse problem are shown to be ill posed a question is thus raised about the validity of these conclusions in the general case various conjugate gradient algorithms coupled with the adjoint state finite element method for computing the gradient of the estimation criterion and with newton s method for determining the optimum step size downgradient are compared a marked improvement in the rate at which these algorithms converge is shown to be achieved by switching from one method to another when the former slows down or fails to converge

time frequency localization operators a geometric phase space approach the author defines a set of operators which localize in both time and frequency these operators are similar to but different from the low pass time limiting operator the singular functions of which are the prolate spheroidal wave functions the author s construction differs from the usual approach in that she treats the time frequency plane as one geometric whole phase space rather than as two separate spaces for disk shaped or ellipse shaped domains in time frequency plane the associated localization operators are remarkably simple their eigenfunctions are hermite functions and the corresponding eigenvalues are given by simple explicit formulas involving the incomplete gamma functions

on model selection criteria in multimodel analysis 1 hydrologic systems are open and complex rendering them prone to multiple conceptualizations and mathematical descriptions there has been a growing tendency to postulate several alternative hydrologic models for a site and use model selection criteria to 1 rank these models 2 eliminate some of them and or 3 weigh and average predictions and statistics generated by multiple models this has led to some debate among hydrogeologists about the merits and demerits of common model selection also known as model discrimination or information criteria such as aic aicc bic and kic and some lack of clarity about the proper interpretation and mathematical representation of each criterion we examine the model selection literature to find that 1 all published rigorous derivations of aic and aicc require that the true model having generated the observational data be in the set of candidate models 2 though bic and kic were originally derived by assuming that such a model is in the set bic has been rederived by cavanaugh and neath 1999 without the need for such an assumption and 3 kic reduces to bic as the number of observations becomes large relative to the number of adjustable model parameters implying that it likewise does not require the existence of a true model in the set of alternatives we explain why kic is the only criterion accounting validly for the likelihood of prior parameter estimates elucidate the unique role that the fisher information matrix plays in kic and demonstrate through an example that it imbues kic with desirable model selection properties not shared by aic aicc or bic our example appears to provide the first comprehensive test of how aic aicc bic and kic weigh and rank alternative models in light of the models predictive performance under cross validation with real hydrologic data

a comparison of four inverse approaches to groundwater flow and transport parameter identification four different methods of parameterizing spatially varying log transmissivities in an inverse approach are compared with respect to prediction accuracy of simulated flow and transport transport parameter estimation is included by two stage feedback optimization in stage one the log transmissivities are estimated by fitting both head and concentration data given initial values of the source concentration and the dispersivities in stage two the source concentration and the dispersivities are estimated by fitting the concentration data with the updated transport parameters final estimates of the log transmissivities are obtained by repeating the optimization of stage one the formulated objective functions are minimized using levenberg marquardt s algorithm the models are applied to synthetic two dimensional transport problems in steady state flow regimes the true log transmissivity fields are generated by the turning bands method thereby incorporating spatial variability the test cases differ in the input variances of the generated fields and with respect to the amount and accuracy of observed transmissivity data

role of the calibration process in reducing model predictive error 1 an equation is derived through which the variance of predictive error of a calibrated model can be calculated this equation has two terms the first term represents the contribution to predictive error variance that results from an inability of the calibration process to capture all of the parameterization detail necessary for the making of an accurate prediction if a model is uncalibrated with parameter values being supplied solely through outside information this is the only term required the second term represents the contribution to predictive error variance arising from measurement noise in an overdetermined system such as that which may be obtained through parameter lumping e g through the introduction of a spatial zonation scheme this is the only term required it is shown however that parameter lumping is a form of implicit regularization and that ignoring the implied first term of the predictive error variance equation can potentially lead to underestimation of predictive error variance a model s role as a predictor of environmental behavior can be enhanced if it is calibrated in such a way as to reduce the variance of those predictions which it is required to make it is shown that in some circumstances this can be accomplished through overfitting against historical field data it can also be accomplished by giving greater weight to those measurements which carry the greatest information content with respect to a required prediction this suggests that a departure may be necessary from the custom of using a single calibrated model for the making of many different predictions instead model calibration may need to be repeated many times so that in each case the calibration process is optimized for the making of a specific model prediction

on rank one perturbations of normal operators this paper is concerned with operators on hilbert space of the form t d u v where d is a diagonalizable normal operator and u v is a rank one operator it is shown that if t c1 and the vectors u and v have fourier coefficients n n 1 and n n 1 with respect to an orthonormal basis that diagonalizes d that satisfy n 1 n 2 3 n 2 3 then t has a nontrivial hyperinvariant subspace this partially answers an open question of at least 30 years duration

toeplitz operators and localization operators we show that for any localization operator on the fock space with polynomial window there exists a constant coecient linear partial dierential operator d such that the localization operator with symbol f coincides with the toeplitz operator with symbol df an analogous result also holds in the context of bergman spaces on bounded symmetric domains this verifies a recent conjecture of coburn and simplifies and generalizes recent results of lo

completeness and spectral synthesis of nonselfadjoint one dimensional perturbations of selfadjoint operators abstract we study spectral properties of nonselfadjoint rank one perturbations of compact selfadjoint operators the problems under consideration include completeness of eigenvectors relations between completeness of the perturbed operator and its adjoint and the spectral synthesis problem we obtain new criteria for completeness and spectral synthesis in this class as well as a series of counterexamples which show that the spectral structure of rank one perturbations is in general unexpectedly rich and complicated a parallel spectral theory is developed for one dimensional singular perturbations of unbounded selfadjoint operators our approach is based on a functional model for this class which translates the properties of operators to completeness problems for systems of reproducing kernels and their biorthogonals in some spaces of analytic entire functions

on the essential commutant of the toeplitz algebra on the bergman space let tt be the c c algebra generated by the toeplitz operators tf f l b dv tf f l b dv on the bergman space of the unit ball we show that the essential commutant of tt equals tg g vobdd k tg g vobdd k where vobddvobdd is the collection of bounded functions of vanishing oscillation on b and kk denotes the collection of compact operators on la2 b dv

compact toeplitz operators on the segal bargmann space abstract in this note we show that miao and zheng s characterization of compact operators on the bergman space of the unit disk that are finite sums of finite products of toeplitz operators with each one of the symbols belonging to bt also holds for the segal bargmann space of c n

inverse groundwater modeling with emphasis on model parameterization 1 this study develops an inverse method aiming to circumvent the subjective decision regarding model parameterization and complexity in inverse groundwater modeling the number of parameters is included as a decision variable along with parameter values a parameterization based on b spline surfaces bss is selected to approximate transmissivity and genetic algorithms were selected to perform error minimization a transform based on linear least squares lls is developed so that different parameterizations may be combined by standard genetic algorithm operators first three applications with isotropic anisotropic and zoned aquifer parameters are examined in a single objective optimization problem and the estimated transmissivity is found to be near the true one interestingly in the anisotropic case the algorithm converged to a solution with an anisotropic distribution of control points next a single objective optimization with regularization penalizing complex models is considered and last the problem is expressed in a multiobjective optimization framework moo where the goals are simultaneous minimization of calibration error and model complexity the result of moo is a pareto set of potential solutions where the user can examine the tradeoffs between calibration error and model complexity and select the most suitable model by comparing calibration with prediction errors it appears that the most promising models are the ones near a region where the rate of decrease of calibration error as model complexity increases drops bend of error curve this is a useful result of practical interest in real inverse modeling applications

identification of parameter structure in groundwater inverse problem this paper develops an optimization method that can be used to identify parameter structure as well as its values in connection with the inverse problem in groundwater modeling the complexity of the parameter structure is determined by the quantity and quality of observation data output least square error is used as the criterion of identification sensitivity coefficients at finite element nodes are evaluated by the variational method the problem of identifying the parameter structure is formulated as a problem of combinatorial optimization a new parameterization technique is presented for describing the parameter structure it can represent various distributions of parameters from piecewise constant to continuous variability based on this representation the combinatorial optimization problem for determining the parameter structure is then solved by a simple one at a time searching procedure numerical experimentations are used to demonstrate the efficiency and practicality of the proposed method

bmo in the bergman metric on bounded symmetric domains for bounded symmetric domains in cn a notion of bounded mean oscillation in terms of the bergman metric is introduced it is shown that for in l2 dv is in bmo if and only if the densely defined operator m p m p pm on l2 dv is bounded here m is multiplication by and p is the bergman projection with range the bergman subspace h2 dv la2 dv of holomorphic functions in l2 dv an analogous characterization of compactness for m p is provided by functions of vanishing mean oscillation at the boundary of

simultaneous estimation of transmissivity values and zonation the extended kalmanfilter ekf has long been recognized as a powerful yet computationally intensive methodology for stochastic parameter estimation three improvements to traditional algorithms are presented and applied to heterogeneous transmissivity estimation first the costly ekf covariance updates are replaced by more efficient approximations second the zonation structure of the distributed parameterfield being estimated is dynamically determined and refined using a partitional clustering algorithm third a new method of mergingfirst and second moments of randomfields that have heterogeneous statistics is introduced we apply this method called random field union as an alternative to conventional randomfield averaging for the systematic shrinking of covariance matrices as the dimensionality of the parameter space is reduced the effects of these three improvements are examined in applications to steady state groundwaterflow test problems we show that thefirst and second improvements reduce the computational time requirements dramatically while the second and third can improve the accuracy and stability of the results the resulting integrated method is successfully applied to a larger more realistic calibration test case under steady and cyclostationary flow conditions similar to regular seasonalfluctuations whenflow is steady the method can be viewed as iterative whenflow is transient the method is fully recursive

ergodic control of switching diffusions we study the ergodic control problem of switching diffusions representing a typical hybrid system that arises in numerous applications such as fault tolerant control systems flexible manufacturing systems etc under fairly general conditions we establish the existence of a stable nonrandomized markov policy which almost surely minimizes the pathwise long run average cost we then study the corresponding hamilton jacobi bellman hjb equation and establish the existence of a unique solution in a certain class using this we characterize the optimal policy as a minimizing selector of the hamiltonian associated with the hjb equations as an example we apply the results to a failure prone manufacturing system and obtain closed form solutions for the optimal policy

efficient regularization and uncertainty analysis using a global optimization methodology 1 the concept of the pareto front has received considerable attention in the model calibration literature particularly in conjunction with global search optimizers that have been developed for use in contexts where objective function surfaces are pitted with local optima or characterized by multiple broad regions of attraction in parameter space in this paper use of the pareto concept in such calibration contexts is extended to include regularization and model predictive uncertainty analysis both of these processes can be formulated as constrained optimization problems in which a trade off is analyzed between a set of constraints on model parameters on the one hand and maximization minimization of one or a number of model outputs of interest on the other hand in both cases the optimal trade off point though being calculable on a theoretical basis for synthetic cases must be chosen subjectively when working with real world models two cases are presented to illustrate the methodology one a synthetic groundwater model and the other a real world surface water model

a comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow this paper describes the first major attempt to compare seven different inverse approaches for identifying aquifer transmissivity the ultimate objective was to determine which of several geostatistical inverse techniques is better suited for making probabilistic forecasts of the potential transport of solutes in an aquifer where spatial variability and uncertainty in hydrogeologic properties are significant seven geostatistical methods fast fourier transform ff fractal simulation fs linearized cokriging lc linearized semianalytical ls maximum likelihood ml pilot point pp and sequential self calibration ss were compared on four synthetic data sets each data set had specific features meeting or not classical assumptions about stationarity amenability to a geostatistical description etc the comparison of the outcome of the methods is based on the prediction of travel times and travel paths taken by conservative solutes migrating in the aquifer for a distance of 5 km four of the methods ls ml pp and ss were identified as being approximately equivalent for the specific problems considered the magnitude of the variance of the transmissivity fields which went as high as 10 times the generally accepted range for linearized approaches was not a problem for the linearized methods when applied to stationary fields that is their inverse solutions and travel time predictions were as accurate as those of the nonlinear methods nonstationarity of the true transmissivity field or the presence of anomalies such as high permeability fracture zones was however more of a problem for the linearized methods the importance of the proper selection of the semivariogram of the log10 t field or the ability of the method to optimize this variogram iteratively was found to have a significant impact on the accuracy and precision of the travel time predictions use of additional transient information from pumping tests did not result in major changes in the outcome while the methods differ in their underlying theory and the codes developed to implement the theories were limited to varying degrees the most important factor for achieving a successful solution was the time and experience devoted by the user of the method

toeplitz operators with bmo symbols on the segal bargmann space we show that zorboska s criterion for compactness of toeplitz operators with bmo 1 symbols on the bergman space of the unit disc holds by a different proof for the segal bargmann space of gaussian square integrable entire functions on n we establish some basic properties of bmot p for p 1 and complete the characterization of bounded and compact toeplitz operators with bmo 1 symbols via the bargmann isometry and results of lo and englis we also give a compactness criterion for the gabor daubechies windowed fourier localization operators on l 2 n dv when the symbol is in a bmo 1 sobolev type space finally we discuss examples of the compactness criterion and counterexamples to the unrestricted application of this criterion for the compactness of toeplitz operators

invariant subspaces for certain finite rank perturbations of diagonal operators abstract suppose that e k is an orthonormal basis for a separable infinite dimensional hilbert space h let d be a diagonal operator with respect to the orthonormal basis e k that is d k 1 k e k e k where k is a bounded sequence of complex numbers let t d u 1 v 1 u n v n improving a result of foias et al 2007 3 we show that if the vectors u 1 u n and v 1 v n satisfy an l 1 condition with respect to the orthonormal basis e k and if t is not a scalar multiple of the identity operator then t has a non trivial hyperinvariant subspace

localization and the toeplitz algebra on the bergman space abstract let t f denote the toeplitz operator with symbol function f on the bergman space l a 2 b d v of the unit ball in c n it is a natural problem in the theory of toeplitz operators to determine the norm closure of the set t f f l b d v in b l a 2 b d v we show that the norm closure of t f f l b d v actually coincides with the toeplitz algebra t i e the c algebra generated by t f f l b d v a key ingredient in the proof is the class of weakly localized operators recently introduced by isralowitz mitkovski and wick our approach simultaneously gives us the somewhat surprising result that t also coincides with the c algebra generated by the class of weakly localized operators

aquifer parameter identification with optimum dimension in parameterization this paper presents a systematic procedure whereby the functional coefficients imbedded in a two dimensional partial differential equation which governs unsteady groundwater flow are optimally identified the coefficients to be identified are transmissivities which vary spatially finite elements are used to represent the unknown transmissivity function parametrically in terms of nodal values over a suitable discretization of a flow region a modified gauss newton algorithm is used for parameter optimization covariance analysis is used to estimate the reliability of the estimated parameters as the dimension of the unknown parameter increases the modeling error represented by a least squares criterion will generally decrease but errors in data would be propagated to a greater degree into the estimated parameters thus reducing the reliability of estimation the reliability of the estimated parameters is characterized by a norm of the covariance matrix this information is used for the determination of the optimum dimension in parameterization

global local optimization for parameter structure identification in three dimensional groundwater modeling 1 this paper develops two global local optimization methods for three dimensional parameter structure identification in groundwater modeling parameter structure identification is formulated in terms of solving a generalized inverse problem which allows for a determination of an appropriate level of parameter structure complexity and the identification of its pattern and the associated parameter values the structure complexity is determined by calculating the parameter structure error measured in the prediction space or management space while the structure pattern and the associated parameter values are identified simultaneously by minimizing the fitting residual measured in the observation space the former requires solving a continuous maximum minimum max min optimization problem and the latter requires solving a combinatorial optimization problem in this paper voronoi zonation is used to parameterize the unknown distributed parameter for each given level of structure complexity a sequential global local optimization method which consists of a genetic algorithm a quasi newton method and local search is developed to solve the combinatorial problem in addition the continuous max min problem is transformed to a hierarchical optimization problem and solved by a hybrid global local method sensitivities of state variables to the unknown parameters are calculated by the sensitivity equations the validity and applicability of the proposed methodology are demonstrated by numerical experiments we have shown that the choice of an objective function in model application impacts the determination of the parameter structure complexity

localization and compactness of operators on fock spaces abstract for 0 p let f p be the fock space induced by a weight function satisfying d d c 0 in this paper given p 0 1 we introduce the concept of weakly localized operators on f p we characterize the compact operators in the algebra generated by weakly localized operators as an application for 0 p we prove that an operator t in the algebra generated by bounded toeplitz operators with bmo symbols is compact on f p if and only if its berezin transform satisfies certain vanishing property at in the classical fock space we extend the axler zheng condition on linear operators t which ensures t is compact on f p for all possible 0 p

localization and berezin transform on the fock space abstract we introduce the notion of sufficiently localized operators on the fock space we show that if a is in the c algebra generated by the class of sufficiently localized operators then a is compact if and only if its berezin transform vanishes at infinity moreover we show that this class contains many familiar operators including all the toeplitz operators with bounded symbols

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

turnpike sets and their analysis in stochastic production planning problems this paper considers optimal infinite horizon stochastic production planning problems with capacity and demand to be finite state markov chains the existence of the optimal feedback control is shown with the aid of viscosity solutions to the dynamic programming equations turnpike set concepts are introduced to characterize the optimal inventory levels it is proved that the turnpike set is an attractor set for the optimal trajectories provided that the capacity is assumed to be fixed at a level exceeding the maximum possible demand conditions under which the optimal trajectories enter the convex closure of the set in finite time are given the structure of turnpike sets is analyzed last but not least it is shown that the turnpike sets exhibit a monotone property with respect to capacity and demand it turns out that the monotonicity property helps in solving the optimal production problem numerically and in some cases analytically

a reassessment of the groundwater inverse problem this paper presents a functional formulation of the groundwaterflow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a possibly infinite basis function expansion with random coefficients the unknown parameter function is related to the measurements used for estimation by a forward operator which describes the measurement process in the particular case considered here the parameter of interest is the large scale log hydraulic conductivity the measurements are point values of log conductivity and piezometric head and the forward operator is derived from an upscaled groundwaterflow equation the inverse algorithm seeks the most probable or maximum a posteriori estimate of the unknown parameter function when the measurement errors and parameter function are gaussian and independent the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness of fit and prior terms when the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate this constraint tends to make the problem well posed by limiting the range of admissible solutions the gaussian maximum a posteriori problem may be solved with variational methods using functional generalizations of gauss newton or gradient based search techniques several popular groundwater inverse algorithms are either special cases of or variants on the functional maximum a posteriori algorithm these algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use linear versus nonlinear the accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a bayesian extension of the cramer rao lower bound on the estimation error covariance this bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements

exact sensitivity matrix and influence of the number of pilot points in the geostatistical inversion of moment equations of groundwater flow received 3 august 2009 revised 12 july 2010 accepted 21 july 2010 published 4 november 2010 1 we present novel equations for the exact sensitivity matrix of the ensemble mean hydraulic head under steady state groundwater flow conditions these equations are embedded in a geostatistical inverse procedure to condition approximations of stochastic moment equations of flow on measured hydraulic conductivities and heads our formulation allows considerable improvement of the methodology proposed by hernandez et al 2003 2006 and renders the inversion of moment equations feasible for a large number of unknown hydraulic parameters the spatial distribution of the natural logarithm y of conductivity is parameterized within the pilot points framework whereas prior values of y at pilot points are obtained by a variant of kriging posterior estimates at pilot points are obtained through a maximum likelihood fit of computed to measured heads the maximum likelihood function also includes a regularization term by means of a synthetic example and upon adopting formal model information criteria we explore the influence of 1 the number of pilot points and 2 the order of approximation of the governing mean flow equation on our ability to properly estimate the log conductivity and head fields and identify the relative weight of the regularization term and the parameters of the underlying y variogram we find that none of the adopted information criteria can identify the optimum number of pilot points and the plausibility weight and variogram parameters values can be determined by the kashyap s bayesian measure

brief paper a new approach to linear regression with multivariate splines a new methodology for creating highly accurate static nonlinear maps from scattered multivariate data is presented this new methodology uses the b form polynomials of multivariate simplex splines in a new linear regression scheme this allows the use of standard parameter estimation techniques for estimating the b coefficients of the multivariate simplex splines we present a generalized least squares estimator for the b coefficients and show how the estimated b coefficient variances lead to a new model quality assessment measure in the form of the b coefficient variance surface the new modeling methodology is demonstrated on a nonlinear scattered bivariate dataset

localization carleson measure and bmo toeplitz operators on the bergman space abstract in this paper we prove that a toeplitz operator with complex borel measure symbol whose total variation is carleson is weakly localized on the bergman space we define strongly localized and sufficiently localized operators on the bergman space and show that they are also weakly localized finally we show that bounded toeplitz operators with b m o 1 symbols are strongly and therefore also weakly localized and that the bergman space toeplitz algebra has yet another characterization as the c algebra generated by this class of operators

spectral decomposability of rank one perturbations of normal operators abstract this paper is a continuation of the study by foias jung ko and pearcy 2007 4 and foias jung ko and pearcy 2008 5 of rank one perturbations of diagonalizable normal operators in foias jung ko and pearcy 2007 4 we showed that there is a large class of such operators each of which has a nontrivial hyperinvariant subspace and in foias jung ko and pearcy 2008 5 we proved that the commutant of each of these rank one perturbations is abelian in this paper we show that the operators considered in foias jung ko and pearcy 2007 4 have more structure namely that they are decomposable operators in the sense of colojoar and foias 1968 1

toeplitz operators on the segal bargmann space in this paper we give a complete characterization of those functions on 2n dimensional euclidean space for which the berezin toeplitz quantizations admit a symbol calculus modulo the compact operators the functions in question are characterized by a condition of small oscillation at infinity

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