ternary 2n point lagrange subdivision schemes ternary 2n point lagrange subdivision schemes with a tension parameter are analyzed it is shown that the resulting curves are c 3 c 4 and c 5 continuous for the values of n 2 3 and 4 respectively the role of the tension parameter in subdivision scheme is demonstrated by a few examples

smooth surface interpolation to scattered data using interpolatory subdivision algorithms in this paper a smooth interpolatory subdivision algorithm for the generation of interpolatory surfaces gc1 over arbitrary triangulations is constructed and its convergence properties over nonuniform triangulations studied an immediate application of this algorithm to surface interpolation to scattered data in rn n 3 is also studied for uniform data this method is a generalization of the analyses for univariate subdivision algorithms and for nonuniform data an extraordinary point analysis is proposed and a local subdivision matrix analysis presented it is proved that the subdivision algorithm produces smooth surfaces over arbitrary networks provided the shape parameters of the algorithm are kept within an appropriate range some error estimates for both uniform and nonuniform triangulations are also investigated finally three graphical examples of surface interpolations over nonuniform data are given to show the smoothing interpolating process of the algorithm

on spacelike hypersurfaces with constant mean curvature in the de sitter space soit n n n 1 c g une variete de lorentz a n 1 dimensions de courbure constante positive c soit m une hypersurface du genre espace complete a courbure moyenne constante h dans n on suppose que 1 h c 1 2 quand n 2 2 h 2 n 1 c 1 2 n quand n 3 alors m est totalement ombilicale

convexity preserving interpolatory subdivision with conic precision the paper is concerned with the problem of shape preserving interpolatory subdivision for arbitrarily spaced planar input data an efficient non linear subdivision algorithm reproducing conic sections and respecting the convexity properties of the initial data is here presented significant numerical examples are included to illustrate the effectiveness of the proposed method and the smoothness of the limit curves

generalized self similarity abstract we prove the existence of l pfunctions satisfying a kind of self similarity condition this is achieved by solving a functional equation by means of the construction of a contractive operator on an appropriate functional space the solution a fixed point of the operator can be obtained by an iterative process making this model very suitable to use in applications such as fractal image and signal compression on the other hand this generalized self similarity equation includes matrix refinement equations of the typef x ckf ax k which are central in the construction of wavelets and multiwavelets the results of this paper will therefore yield conditions for the existence of l p refinable functions in a very general setting

on interpolatory subdivision symbol formulation and parameter convergence intervals abstract this paper is concerned with general symmetric 2 n point interpolatory subdivision scheme iss with polynomial reproduction of arbitrary order m n an explicit formulation is derived for the corresponding refinement symbol and convergence intervals are obtained for the one parameter case the left hand endpoints of which improve on previous such lower parameter convergence bounds

polynomial based non uniform interpolatory subdivision with features control starting from a well known construction of polynomial based interpolatory 4 point schemes in this paper we present an original affine combination of quadratic polynomial samples that leads to a non uniform 4 point scheme with edge parameters this blending type formulation is then further generalized to provide a powerful subdivision algorithm that combines the fairing curve of a non uniform refinement with the advantages of a shape controlled interpolation method and an arbitrary point insertion rule the result is a non uniform interpolatory 4 point scheme that is unique in combining a number of distinctive properties in fact it generates visually pleasing limit curves where special features ranging from cusps and flat edges to point edge tension effects may be included without creating undesired undulations moreover such a scheme is capable of inserting new points at any positions of existing intervals so that the most convenient parameter values may be chosen as well as the intervals for insertion such a fully flexible curve scheme is a fundamental step towards the construction of high quality interpolatory subdivision surfaces with features control

shape controlled interpolatory ternary subdivision ternary subdivision schemes compare favorably with their binary analogues because they are able to generate limit functions with the same or higher smoothness but smaller support in this work we consider the two issues of local tension control and conics reproduction in univariate interpolating ternary refinements we show that both these features can be included in a unique interpolating 4 point subdivision method by means of non stationary insertion rules that do not affect the improved smoothness and locality of ternary schemes this is realized by exploiting local shape parameters associated with the initial polyline edges

exact interpolation and iterative subdivision schemes we examine the circumstances under which a discrete time signal can be exactly interpolated given only every mth sample after pointing out the connection between designing an m fold interpolator and the construction of an m channel perfect reconstruction filter bank we derive necessary and sufficient conditions on the signal under which exact interpolation is possible bandlimited signals are one obvious example but numerous others exist we examine these and show how the interpolators may be constructed a main application is to iterative interpolation schemes used for the efficient generation of smooth curves we show that conventional bandlimited interpolators are not suitable in this context a better criterion is to use interpolators that are exact for polynomial functions we demonstrate that these interpolators converge when iterated and show how these may be designed for any polynomial degree n and any interpolation factor m this makes it possible to design interpolators for iterative schemes to make best use of the resolution available in a given display medium

how data dependent is a nonlinear subdivision scheme a case study based on convexity preserving subdivision the regularity of the limit function of a linear subdivision scheme is essentially irrelevant to the initial data how data dependent then is the regularity of the limit of a nonlinear subdivision scheme the answer is the most obvious it depends in this paper we prove that the nonlinear convexity preserving subdivision scheme developed independently by floater and micchelli m s floater and c a micchelli approximation theory marcel dekker new york 1998 pp 209 224 and kuijt and van damme f kuijt and r van damme constr approx 14 1998 pp 609 630 exhibits a rather strong nonlinear data dependent behavior for any nu in 1 2 there exists initial convex data such that the critical holder regularity of the limit curve is exactly nu we also show that the limit function of any initial data always has holder regularity less than 2 unless if restricted to a subset of the domain at which the initial data is sampled from the convex branch of a rational polynomial of degree 2 over degree 1 this result stands in contrast to what is reported in several recent publications on nonlinear subdivision schemes i ur rahman i drori v c stodden d l donoho and p schr oder siam j multiscale modeling and simulation submitted 2005 g xie and t p y yu constr approx 22 2005 pp 219 254 g xie and t p y yu advances in constructive approximation 2004 pp 519 533 i daubechies o runborg and w sweldens constr approx 3 2004 pp 399 463 t p y yu cutting corners on the sphere preprint available at http www rpi edu yut papers cuttingcorners pdf 2005 in which various families of nonlinear subdivision schemes are either proved or empirically observed to have rather weak nonlinearity in the sense that they produce limit curves with smoothness insensitive to initial data

c inf interpolation of discrete fields on regular and irregular grids the article proposes a simple c interpolation of discretized lattice fields on regular and irregular grids the method is based on localized c but not analytic basis functions which vanish outside an open set region of influence as a result the interpolating fields at a point depend exclusively on the nodal values within the region of influence the method can be applied to generic fields whose support is a limited set of n dimensional space starting from discretized values given on regular or irregular grids particular attention is focused on the interpolation of cfd computed velocity fields that give rise to lagrangian chaos

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

efficient evaluation of subdivision schemes with polynomial reproduction property in this paper we present an efficient framework for the evaluation of subdivision schemes with polynomial reproduction property for all interested rational parameters between 0 and 1 with the same denominator their exact limit positions on the subdivision curve can be obtained by solving a system of linear equations when the framework is applied to binary and ternary 4 point interpolatory subdivision schemes we find that the corresponding coefficient matrices are strictly diagonally dominant and so the evaluation processes are robust for any individual irrational parameters between 0 and 1 its approximate value is computed by a recursive algorithm which can attain an arbitrary error bound for surface schemes generalizing univariate subdivision schemes with polynomial reproduction property exact evaluation methods can also be derived by combining stam s method with that of this paper the method is applicable to all subdivision schemes with polynomial reproduction it performs exact evaluation at rational parameters and approximate evaluation at other arbitrary parameters with tolerance control it is efficient and robust for the presented schemes with corresponding coefficient matrix being strictly and diagonally dominant it can also evaluate derivatives under the same framework extension of the method to surface cases is straightforward

from approximating subdivision schemes for exponential splines to high performance interpolating algorithms in this work we construct three novel families of approximating subdivision schemes that generate piecewise exponential polynomials and we show how to convert these into interpolating schemes of great interest in curve design for their ability to reproduce important analytical shapes and to provide highly smooth limit curves with a controllable tension in particular throughout this paper we will focus on the derivation of 6 point interpolating schemes that turn out to be unique in combining vital ingredients like c 2 continuity simplicity of definition ease of implementation user independency tension control and ability to reproduce salient trigonometric and transcendental curves

general empirical bayes wavelet methods and exactly adaptive minimax estimation in many statistical problems stochastic signals can be represented as a sequence of noisy wavelet coefficients in this paper we develop general empirical bayes methods for the estimation of true signal our estimators approximate certain oracle separable rules and achieve adaptation to ideal risks and exact minimax risks in broad collections of classes of signals in particular our estimators are uniformly adaptive to the minimum risk of separable estimators and the exact minimax risks simultaneously in besov balls of all smoothness and shape indices and they are uniformly superefficient in convergence rates in all compact sets in besov spaces with a finite secondary shape parameter furthermore in classes nested between besov balls of the same smoothness index our estimators dominate threshold and james stein estimators within an infinitesimal fraction of the minimax risks more general block empirical bayes estimators are developed both white noise with drift and nonparametric regression are considered

high level continuity for coordinate generation with precise controls abstract the development of precisely controlled coordinate generation techniques is continued from a first study to include the higher order smoothness which is necessary for three dimensional applications in the first study the controls came from the use of local piecewise linear interpolants in the general multisurface transformation the consequent integration therein resulted in coordinates with continuity up to first derivatives and with the capability to prescribe uniformity either locally or globally for the family of transverse coordinate curves the admission of uniformity placed a constraint upon the general interpolants which was trivially satisfied in the piecewise linear case with smoother piecewise constructions the constraint is used herein along with a requirement for coordinate curves to have the most general possible curvature properties the result is a class of coordinate transformations with continuity extended up to higher derivatives that retains the precise local controls displayed in the first study and that can be used in three or more dimensions

compactly supported fundamental functions for spline interpolation in this paper various ways of constructing locally supported fundamental splines leading to highly accurate local interpolation schemes are proposed and analyzed

on compactly supported spline wavelets and a duality principle let v 1 v 0 v 1 be a multiresolution analysis of l 2 generated by the mth order b spline n m x we exhibit a compactly supported basic wavelet m x that generates the corresponding orthogonal complementary wavelet subspaces w 1 w 0 w 1 consequently the two finite sequences that describe the two scale relations of n m x and m x in terms of n m 2x j j z yield an efficient reconstruction algorithm to give an efficient wavelet decomposition algorithm based on these two finite sequences we derive a duality principle which also happens to yield the dual bases n m x j and 59 m x j relative to n m x j and m x j respectively

convexity preservation of five point binary subdivision scheme with a parameter a five point binary subdivision scheme with a parameter u is presented the generating polynomial method and the holder exponent are used to investigate the uniform convergence and c k continuity of this subdivision scheme where k depends on the choice of the parameter u moreover the conditions on the initial points are discussed for the given limit curve to be convexity preserving and an example is given to illustrate our conclusion

a combined ternary 4 point subdivision scheme combined binary 6 point interpolating and approximating subdivision scheme with tension parameters is analyzed it is shown that the resulting curves are c1 c2 interpolating continuous and c1 c2 c3 approximating continuous for different values of tension parameters the role of the tension parameters in subdivision scheme are illustrated using a few examples

wavelets in wandering subspaces mallat s construction via a multiresolution approximation of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions the connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in hilbert space is exploited in the general setting an example of multiresolution approximation generated by cardinal hermite b splines is constructed

refinable functions functionals and iterated function systems invariant measures of iterated function systems and refinable functions are mathematical tools of vast usage that although usually considered in different contexts are deeply linked we outline these links in particular with respect to existence and regularity of these objects

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

wamr an adaptive wavelet method for the simulation of compressible reacting flow part i accuracy and efficiency of algorithm abstract the wavelet adaptive multiresolution representation wamr method provides a robust method for controlling spatial grid adaptation fine grid spacing in regions where a solution varies greatly i e near steep gradients or near singularities and a much coarser grid where the solution varies slowly subsequently a wide range of spatial scales often demanded in challenging continuum physics problems can be efficiently captured furthermore the wavelet transform provides a direct measure of local error at each collocation point effectively producing automatically verified solutions the method is applied to the solution of unsteady compressible reactive flow equations and includes detailed diffusive transport and chemical kinetics models accuracy and performance of the method are examined on several test problems the sparse grids produced by the wamr method exhibit an impressive compression of the solution reducing the number of collocation points used by factors of many orders of magnitude when compared to uniform grids of equivalent resolution

a general framework for local interpolation we present a general framework for the construction of local interpolation methods with a given approximation order some applications to multivariate spline spaces are presented

m band compactly supported orthogonal symmetric interpolating scaling functions in many applications wavelets are usually expected to have the following properties compact support orthogonality linear phase regularity and interpolation to construct such wavelets it is crucial designing scaling functions with the above properties in two and three band cases except for the haar functions there exists no scaling function with the above five properties in m band case m spl ges 4 more free degrees available in design enable us to construct such scaling functions a novel approach to designing such scaling functions is proposed first we extend the two band dubuc 1986 filters to the m band case next the m band fir regular symmetric interpolating scaling filters are parameterized and then m band fir regular orthogonal symmetric interpolating scaling filters osisfs are designed via optimal selection of parameters finally two family of four band and five band osisfs and scaling functions are developed and their smoothness are estimated

non uniform interpolatory subdivision surface abstract this paper presents an interpolatory subdivision scheme with non uniform parametrization for arbitrary polygon meshes with arbitrary manifold topology this is the first attempt to generalize the non uniform four point interpolatory curve subdivision to surface with extraordinary points the scheme is constructed from the inspiration of the relation between the non uniform four point interpolatory subdivision scheme and the non uniform b spline refinement rule numerical examples and comparisons with the uniform interpolatory subdivision schemes indicate that the quality of the limit surface can be improved by using non uniform parameter values for non uniform initial data

optimal local spline interpolants abstract a constructive proof is given of the existence of a local spline interpolant which also approximates optimally in the sense that its associated operator reproduces polynomials of maximal order first it is shown that such an interpolant does not exist for orders higher than the linear case if the partition points of the appropriate spline space coincide with the given interpolation points next in the main result the desired existence of an optimal local spline interpolant for all orders is proved by increasing in a specified manner the set of partition points although our interpolant reproduces a more restricted function space than its quasi interpolant counterpart constructed by de boor and fix 1 it has the advantage of interpolating every real function at a given set of points finally we do some explicit calculations in the quadratic case

a family of ternary subdivision schemes for curves a family of ternary 3 point subdivision schemes is introduced and has a degree of smoothness c1 continuity a family of ternary 4 point subdivision schemes is derived and has degrees of smoothness c1 and c2 continuities also a ternary 4 point approximating subdivision scheme is proposed that generates the limiting curve of c3 continuity the eigenvalue analysis and generating function formalism are used to analyze the continuity properties of these subdivision schemes also these subdivision schemes are compared with the established subdivision schemes

continuous and differentiable multidimensional iterative interpolation abstract we describe a multidimensional iterative interpolation process some examples of the two dimensional process are presented we give a sufficient condition for the continuity and the differentiability of the interpolation functions all the examples are continuous and differentiable

efficient algorithms for deciding the type of growth of products of integer matrices for a given finite set sigma of matrices with nonnegative integer entries we study the growth with t of max parallel to a 1 a t parallel to a i epsilon sigma we show how to determine in polynomial time whether this growth is bounded polynomial or exponential and we characterize all possible behaviors c 2007 elsevier inc all rights reserved

applications of optimally local interpolation to interpolatory approximants and compactly supported wavelets the objective of this paper is to introduce a general scheme for the construction of interpolatory approximation formulas and compactly supported wavelets by using spline functions with arbitrary nonuniform knots both construction procedures are based on certain optimally local interpolatory fundamental spline functions which are not required to possess any approximation property

sets of matrices all infinite products of which converge an infinite product iit lmi of matrices converges on the right if limi m mi exists a set z ai i l of n x n matrices is called an rcp set rightconvergent product set if all infinite products with each element drawn from z converge such sets of matrices arise in constructing self similar objects like von koch s snowflake curve in various interpolation schemes in constructing wavelets of compact support and in studying nonhomogeneous markov chains this paper gives necessary conditions and also some sufficient conditions for a set x to be an rcp set these are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices necessary and sufficient conditions are given for a finite set z to be an rcp set having a limit function m d rit lad where d d d which is a continuous function on the space of all sequences d with the sequence topology finite rcp sets of column stochastic matrices are completely characterized some results are given on the problem of algorithmically deciding if a given set x is an rcp set

multidimensional interpolatory subdivision schemes this paper presents a general construction of multidimensional interpolatory subdivision schemes in particular we provide a concrete method for the construction of bivariate interpolatory subdivision schemes of increasing smoothness by finding an appropriate mask to convolve with the mask of a three direction box spline br r r of equal multiplicities the resulting mask for the interpolatory subdivision exhibits all the symmetries of the three direction box spline and with this increased symmetry comes increased smoothness several examples are computed for r 2 8 regularity criteria in terms of the refinement mask are established and applied to the examples to estimate their smoothness

smoothness equivalence properties of manifold valued data subdivision schemes based on the projection approach interpolation of manifold valued data is a fundamental problem which has applications in many fields the linear subdivision method is an efficient and well studied method for interpolating or approximating real valued data in a multiresolution fashion a natural way to apply a linear subdivision scheme overline s to interpolate manifold valued data is to first embed the manifold at hand into an euclidean space and construct a projection operator p that maps points from the ambient space to a closest point on the embedded surface and then consider the nonlinear subdivision operator s p circ overline s when applied to symmetric spaces such as s n 1 so n sl n se n g n k the projection method can also be carried out in such a way that the resulting schemes enjoy natural coordinate invariance properties and robust numerical implementations despite such nice features the mathematical analysis of such nonlinear subdivision schemes is in its infancy in this article we attack the so called smoothness equivalence conjecture which asserts that the smoothness property of s is exactly the same as that of overline s we show that in the cases of s n 1 so n and related manifolds we have a proximity condition of the form s overline s y infty lesssim sum i 1 p 1 delta i y infty delta p i y infty where p is the accuracy order of overline s armed with this proximity condition and other known approximation theoretic results we can establish the result that the ho lder smoothness exponent of s is always as high as that of overline s no matter how high the latter is

new characterizations for hyperbolic cylinders in anti de sitter spaces abstract in this paper we study complete maximal spacelike hypersurfaces in anti de sitter space h 1 n 1 with either constant scalar curvature or constant non zero gauss kronecker curvature we characterize the hyperbolic cylinders h m c 1 h n m c 2 1 m n 1 as the only such hypersurfaces with n 1 principal curvatures with the same sign everywhere in particular we prove that a complete maximal spacelike hypersurface in h 1 5 with negative constant gauss kronecker curvature is isometric to h 1 c 1 h 3 c 2

a tetrahedron based subdivision scheme for spatial g 1 curves in this paper we propose a new purely geometrical interpolatory hermite subdivision scheme for generating spatial subdivision curves which starts with a sequence of points and associated unit tangent vectors the newly generated point lies inside a certain tetrahedron which is formed by the given hermite data the method is local and we prove that by iterating this refinement procedure the limit curve is g 1 continuous the additional property of the scheme is that planar data are preserved i e planar subdivision curves are generated for planar initial hermite data and moreover the scheme is circle preserving

a unified framework for interpolating and approximating univariate subdivision in this paper we show that the refinement rules of interpolating and approximating univariate subdivision schemes with odd width masks of finite support can be derived ones from the others by simple operations on the mask coefficients these operations are formalized as multiplication division of the associated generating functions by a proper link polynomial we then apply the proposed result to some families of stationary and non stationary subdivision schemes showing that it also provides a constructive method for the definition of novel refinement algorithms

optimal c 2 two dimensional interpolatory ternary subdivision schemes with two ring stencils for any interpolatory ternary subdivision scheme with two ring stencils for a regular triangular or quadrilateral mesh we show that the critical holder smoothness exponent of its basis function cannot exceed log 3 11 2 18266 where the critical holder smoothness exponent of a function f r 2 r is defined to be v f sup v f lipv on the other hand for both regular triangular and quadrilateral meshes we present several examples of interpolatory ternary subdivision schemes with two ring stencils such that the critical holder smoothness exponents of their basis functions do achieve the optimal smoothness upper bound log 3 11 consequently we obtain optimal smoothest c 2 interpolatory ternary subdivision schemes with two ring stencils for the regular triangular and quadrilateral meshes our computation and analysis of optimal multidimensional subdivision schemes are based on the projection method and the l p norm joint spectral radius

spline prewavelets for non uniform knots in this paper we develop a framework suitable for performing a multiresolution analysis using univariate spline spaces of arbitrary degree and with non uniform knot sequences to this end we show among other things the existence of compactly supported prewavelets and of prewavelets that are globally supported but decay exponentially in each case we obtain a decomposition of a fine spline space as a sum of a coarse spline space plus a spline space spanned by prewavelets

a new characterization of hyperbolic cylinder in anti de sitter space h1n 1 1 abstract by investigating n dimensional complete maximal spacelike hypersurfaces with two distinct principal curvatures in an n 1 dimensional anti de sitter space h 1 n 1 1 we give a new characterization of hyperbolic cylinder h m n m h n m n n m in h 1 n 1 1

new characterizations of hyperbolic cylinders in semi riemannian space forms abstract through the study of the so called linear weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector in a semi riemannian space form q p n p c of index p and constant sectional curvature c 1 0 1 our aim in this paper is establish new characterizations for the hyperbolic cylinders of q p n p c our approach is based on the use of a simons type formula related to an appropriated cheng yau modified operator jointly with the application of suitable generalized maximum principles

initialization and inner product computations of wavelet transform by interpolatory subdivision scheme the initialization of wavelet transforms and the inner product computations of wavelets with their derivatives are very important in many applications in this correspondence the interpolatory subdivision scheme iss is proposed to solve these problems efficiently we introduce a general procedure to compute the exact values of derivatives of the interpolatory fundamental function and then derive a fast recursive algorithm for the realization of the initialization and inner product evaluations error analysis of the algorithm and its comparison with other approaches are discussed numerical experiments demonstrate the high performance of the algorithm

fast implementation of scale space by interpolatory subdivision scheme while the scale space approach has been widely used in computer vision there has been a great interest in fast implementation of scale space filtering we introduce an interpolatory subdivision scheme iss for this purpose in order to extract the geometric features in a scale space representation discrete derivative approximations are usually needed hence a general procedure is also introduced to derive exact formulae for numerical differentiation with respect to this iss then from iss an algorithm is derived for fast approximation of scale space filtering moreover the relationship between the iss and the whittaker shannon sampling theorem and the commonly used spline technique is discussed as an example of the application of iss technique we present some examples on fast implementation of spl lambda spl tau spaces as introduced by gokmen and jain 1997 which encompasses various famous edge detection filters it is shown that the iss technique demonstrates high performance in fast implementation of the scale space filtering and feature extraction

on the complete linear weingarten spacelike hypersurfaces with two distinct principal curvatures in lorentzian space forms abstract we deal with complete linear weingarten spacelike hypersurfaces immersed in a lorentzian space form having two distinct principal curvatures in this setting we show that such a spacelike hypersurface must be isometric to a certain isoparametric hypersurface of the ambient space under suitable restrictions on the values of the mean curvature and of the norm of the traceless part of its second fundamental form our approach is based on the use of a simons type formula related to an appropriated cheng yau modified operator jointly with some generalized maximum principles

lagrange wavelets for signal processing this paper deals with the design of interpolating wavelets based on a variety of lagrange functions combined with novel signal processing techniques for digital imaging halfband lagrange wavelets b spline lagrange wavelets and gaussian lagrange lagrange distributed approximating functional daf wavelets are presented as specific examples of the generalized lagrange wavelets our approach combines the perceptually dependent visual group normalization vgn technique and a softer logic masking slm method these are utilized to rescale the wavelet coefficients remove perceptual redundancy and obtain good visual performance for digital image processing

c1 continuity of the generalized four point scheme abstract we show that the interpolatory four point scheme with tension parameter generates c 1 limit curves if and only if 0 where 0 19273 is the unique real solution of the cubic equation 32 3 4 1 0

a ternary 4 point approximating subdivision scheme in the implementation of subdivision scheme three of the most important issues are smoothness size of support and approximation order our objective is to introduce an improved ternary 4 point approximating subdivision scheme derived from cubic polynomial interpolation which has smaller support and higher smoothness comparing to binary 4 point and 6 point schemes ternary 3 point and 4 point schemes see table 2 the method is easily generalized to ternary 2n 2 point approximating subdivision schemes we choose a ternary scheme because a way to get smaller support is to raise arity and we use polynomial reproduction to get higher approximation order easily 2007 elsevier inc all rights reserved

discrete multiresolution analysis using hermite interpolation biorthogonal multiwavelets we generalize harten s interpolatory multiresolution representation to include hermite interpolation compact hermite interpolation with optimal order accuracy is used in both the decomposition and reconstruction algorithm the resulting multiple basis functions biorthogonal multiwavelets are symmetric or skew symmetric compact and analytic harten s approach has several advantages the multiresolution scheme is inherently discrete nonperiodic boundary conditions are easy to implement and the representation can be extended to nonuniform grids in bounded domains we demonstrate the compression features of the new multiple basis functions by application to several examples

non uniform interpolatory subdivision via splines we present a framework for deriving non uniform interpolatory subdivision algorithms closely related to non uniform spline interpolants families of symmetric non uniform interpolatory 2n point schemes of smoothness c n 1 are presented for n 2 3 4 and even higher order as well as a variety of non uniform 6 point schemes with c 3 continuity

interpolating wavelet collocation method of time dependent maxwell s equations characterization of electrically large optical waveguide discontinuities biorthogonal interpolating wavelets have been applied to electromagnetic field modeling through the wavelet collocation method in time domain yielding a versatile first principle algorithm for the solution of time dependent maxwell s equations with inhomogeneous media the resulting scheme maintains high accuracy while by virtue of its sub gridding capability significant reduction of the computational expenditure has been obtained the proposed method has been applied to the analysis of two dimensional dielectric waveguide discontinuities particularly for the modeling of electrically large optical waveguides where the dimension of the analyzed structure is much larger than the wavelength of the highest frequency content of the transmitted signal the proposed method has been proven to be highly efficient compared to the standard finite difference method

recursive biorthogonal interpolating wavelets and signal adapted interpolating filter banks in this paper by combining the ideas of the recursive wavelets with second generation wavelets a family of recursive biorthogonal interpolating wavelets rbiws is developed the rbiws have simple shape parameter vectors on each level which allows a multichannel decomposition algorithm and provides a flexible structure for designing signal adapted interpolating filter banks in the single level case an efficient approach to design an optimum two channel biorthogonal interpolating filter bank is proposed which maximizes the coding gain under the traditional quantization noise assumption furthermore in the multilevel case using level wise optimization of the shape parameter vectors signal adapted tree structured recursive biorthogonal interpolating filter banks rbifbs are designed which are efficient in computation and can remarkedly improve the coding gain finally numerical results demonstrate the effectiveness of the proposed methods

smooth curve interpolation with generalized conics abstract efficient algorithms for shape preserving approximation to curves and surfaces are very important in shape design and modelling in cad cam systems in this paper a local algorithm using piecewise generalized conic segments is proposed for shape preserving curve interpolation it is proved that there exists a smooth piecewise generalized conic curve which not only interpolates the data points but also preserves the convexity of the data furthermore if the data is strictly convex then the interpolant could be a locally adjustable gc 2 curve provided the curvatures at the data points are well determined it is also shown that the best approximation order is o h 6 an efficient algorithm for the simultaneous computation of points on the curve is derived so that the curve can be easily computed and displayed the numerical complexity of the algorithm for computing n points on the curve is about 2 n multiplications and n additions finally some numerical examples with graphs are provided and comparisons with both quadratic and cubic spline interpolants are also given

the approximation order of four point interpolatory curve subdivision in this paper we derive an approximation property of four point interpolatory curve subdivision based on local cubic polynomial fitting we show that when the scheme is used to generate a limit curve that interpolates given irregularly spaced points sampled from a curve in any space dimension with a bounded fourth derivative and the chosen parameterization is chordal the accuracy is fourth order as the mesh size goes to zero in contrast uniform and centripetal parameterizations yield only second order

solving hyperbolic pdes using interpolating wavelets a method is presented for adaptively solving hyperbolic pdes the method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids the adaptability is performed automatically by thresholding the wavelet coefficients operations such as differentiation and multiplication are fast and simple due to the one to one correspondence between point values and wavelet coefficients in the interpolating basis treatment of boundary conditions is simplified in this sparse point representation spr numerical examples are presented for one and two dimensional problems it is found that the proposed method outperforms a finite difference method on a uniform grid for certain problems in terms of flops

semi regular dubuc deslauriers wavelet tight frames abstract in this paper we construct wavelet tight frames with n vanishing moments for dubuc deslauriers 2 n point semi regular interpolatory subdivision schemes our motivation for this construction is its practical use for further regularity analysis of wide classes of semi regular subdivision our constructive tools are local eigenvalue convergence analysis for semi regular dubuc deslauriers subdivision the unitary extension principle and the generalization of the oblique extension principle to the irregular setting by chui he and stockler this group of authors derives suitable approximation of the inverse gramian for irregular b spline subdivision our main contribution is the derivation of the appropriate approximation of the inverse gramian for the semi regular dubuc deslauriers scaling functions ensuring n vanishing moments of the corresponding framelets

new constructions of nonseparable tight wavelet frames abstract we present two methods for constructing new nonseparable multidimensional tight wavelet frames by combining the ideas of sum of squares representations of nonnegative trigonometric polynomials with the coset sum method of generating nonseparable multidimensional lowpass filters from univariate lowpass filters in effect these methods allow one to select a univariate lowpass filter and generate nonseparable multidimensional tight wavelet frames from it in any dimension n 2 under certain conditions on the input filter which are given explicitly we construct sum of hermitian squares representations for a particular class of trigonometric polynomials f in several variables each related to a coset sum generated lowpass mask in that nonnegativity of f implies the sub qmf condition for in two ways for interpolatory inputs to the coset sum method satisfying the univariate sub qmf condition we find this representation using the fejer riesz lemma and in the general case by writing f x p x where x is a vector of complex exponential functions and p is a constant positive semidefinite matrix that is constructed to reduce the number of generators in this representation the generators of this representation of f may then be used to generate the filters in a tight wavelet frame with lowpass mask several examples of these representations and the corresponding frames are given throughout

on a wavelet based method for the numerical simulation of wave propagation a wavelet based method for the numerical simulation of acoustic and elastic wave propagation is developed using a displacement velocity formulation and treating spatial derivatives with linear operators the wave equations are rewritten as a system of equations whose evolution in time is controlled by first order derivatives the linear operators for spatial derivatives are implemented in wavelet bases using an operator projection technique with nonstandard forms of wavelet transform using a semigroup approach the discretized solution in time can be represented in an explicit recursive form based on taylor expansion of exponential functions of operator matrices the boundary conditions are implemented by augmenting the system of equations with equivalent force terms at the boundaries the wavelet based method is applied to the acoustic wave equation with rigid boundary conditions at both ends in 1 d domain and to the elastic wave equation with a traction free boundary conditions at a free surface in 2 d spatial media the method can be applied directly to media with plane surfaces and surface topography can be included with the aid of distortion of the grid describing the properties of the medium the numerical results are compared with analytic solutions based on the cagniard technique and show high accuracy the wavelet based approach is also demonstrated for complex media including highly varying topography or stochastic heterogeneity with rapid variations in physical parameters these examples indicate the value of the approach as an accurate and stable tool for the simulation of wave propagation in general complex media

from symmetric subdivision masks of hurwitz type to interpolatory subdivision masks in this paper we present a general strategy to deduce a family of interpolatory masks from a symmetric hurwitz non interpolatory one this brings back to a polynomial equation involving the symbol of the non interpolatory scheme we start with the solution of the polynomial equation here proposed tailored for symmetric hurwitz subdivision symbols leads to an efficient procedure for the computation of the coefficients of the corresponding family of interpolatory masks several examples of interpolatory masks associated with classical approximating masks are given

multiscale computation with interpolating wavelets multiresolution analyses based upon interpolets interpolating scaling functions introduced by deslauriers and dubuc are particularly well suited to physical applications because they allowexactrecovery of the multiresolution representation of a function from its sample values on afiniteset of points in space we present a detailed study of the application of wavelet concepts to physical problems expressed in such bases the manuscript describes algorithms for the associated transforms which for properly constructed grids of variable resolution compute correctly without having to introduce extra grid points we demonstrate that for the application of local homogeneous operators in such bases the nonstandard multiply of beylkin coifman and rokhlin also proceeds exactly for inhomogeneous grids of appropriate form to obtain less stringent conditions on the grids we generalize the nonstandard multiply so that communication may proceed between nonadjacent levels the manuscript concludes with timing comparisons against nai x0308 ve algorithms and an illustration of the scale independence of the convergence rate of the conjugate gradient solution of poisson s equation using a simple preconditioning

on nash images of euclidean spaces abstract in this work we characterize the subsets of r n that are images of nash maps f r m r n we prove shiota s conjecture and show that a subset s r n is the image of a nash map f r m r n if and only if s is semialgebraic pure dimensional of dimension d m and there exists an analytic path 0 1 s whose image meets all the connected components of the set of regular points of s two remarkable consequences are the following 1 pure dimensional irreducible semialgebraic sets of dimension d with arc symmetric closure are nash images of r d and 2 semialgebraic sets are projections of irreducible algebraic sets whose connected components are nash diffeomorphic to euclidean spaces

approximation on nash sets with monomial singularities this paper is devoted to the approximation of differentiable semialgebraic functions by nash functions approximation by nash functions is known for semialgebraic functions defined on an affine nash manifold m and here we extend it to functions defined on nash subsets x of m whose singularities are monomial to that end we discuss first finiteness and weak normality for such sets x namely we prove that i x is the union of finitely many open subsets each nash diffeomorphic to a finite union of coordinate linear varieties of an affine space and ii every function on x which is nash on every irreducible component of x extends to a nash function on m then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on nash sets with monomial singularities as a nice consequence we show that m dimensional affine nash manifolds with divisorial corners which are class k semialgebraically diffeomorphic for k m 2 are also nash diffeomorphic

on a nonlinear subdivision scheme avoiding gibbs oscillations and converging towards c s functions with s 1 this paper presents a new nonlinear dyadic subdivision scheme eliminating the gibbs oscillations close to discontinuities its convergence stability and order of approximation are analyzed it is proved that this scheme converges towards limit functions holder continuous with exponent larger than 1 299 numerical estimates provide a holder exponent of 2 438 this subdivision scheme is the first one that simultaneously achieves the control of the gibbs phenomenon and has limit functions with holder exponent larger than 1

interproximate curve subdivision this paper presents a new curve subdivision algorithm called interproximate subdivision for generating curves that interpolate some given vertices and approximate the other vertices by the interproximate subdivision only the vertices specified to be interpolated are fixed and the other vertices are updated at each refinement step the refinement rules are derived to ensure that the eigenvalues of the refinement matrix satisfy the necessary condition of c 2 continuity the interproximate subdivision also contains tension parameters assigned to vertices or edges for shape adjustment compared to the 4 point interpolatory subdivision scheme the interproximate subdivision does not force the new inserted vertices to be interpolated and is thus expected to have improved behavior and compared to the cubic b spline refinement scheme the interproximate subdivision is able to generate curves interpolating user specified vertices in addition the paper also presents two extensions of the interproximate subdivision one automatically adapts the tension parameters locally according to the geometry of the control polygon during the refinement to achieve convexity preservation and the other automatically relaxes the interpolating property of some vertices to achieve better shape behavior

interpolation through an iterative scheme abstract a new method of interpolation is generated through an iterative scheme on dyadic rationals the process is linear local and produces almost twice differentiable functions a fundamental interpolating function is analyzed error formulas are derived the method is applied for curve fitting