sampling and reconstruction of signals in a reproducing kernel subspace of lp rd in this paper we consider sampling and reconstruction of signals in a reproducing kernel subspace of lp rd 1 p associated with an idempotent integral operator whose kernel has certain off diagonal decay and regularity the space of p integrable non uniform splines and the shift invariant spaces generated by finitely many localized functions are our model examples of such reproducing kernel subspaces of lp rd we show that a signal in such reproducing kernel subspaces can be reconstructed in a stable way from its samples taken on a relatively separated set with sufficiently small gap we also study the exponential convergence consistency and the asymptotic pointwise error estimate of the iterative approximation projection algorithm and the iterative frame algorithm for reconstructing a signal in those reproducing kernel spaces from its samples with sufficiently small gap

projective modules over noncommutative tori are multi window gabor frames for modulation spaces in the present investigation we link noncommutative geometry over noncommutative tori with gabor analysis where the first has its roots in operator algebras and the second in time frequency analysis we are therefore in the position to invoke modern methods of operator algebras e g topological stable rank of banach algebras to display the deeper properties of gabor frames furthermore we are able to extend results due to connes and rieffel on projective modules over noncommutative tori to banach algebras which arise in a natural manner in gabor analysis the main goal of this investigation is twofold i an interpretation of projective modules over noncommutative tori in terms of gabor analysis and ii to show that the morita rieffel equivalence between noncommutative tori is the natural framework for the duality theory of gabor frames more concretely we interpret generators of projective modules over noncommutative tori as the gabor atoms of multi window gabor frames for modulation spaces moreover we show that this implies the existence of good multi window gabor frames for modulation spaces with gabor atoms in e g feichtinger s algebra or in schwartz space

newton interpolation in fejer and chebyshev points soit une courbe de jordan dans le plan complexe et soit l ensemble compact borne par soit f une fonction analytique sur l approximation de f sur par un polynome p de degre inferieur a n et qui interpole f en n points de est etudiee

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

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

on the lebesgue constant of barycentric rational interpolation at equidistant nodes recent results reveal that the family of barycentric rational interpolants introduced by floater and hormann is very well suited for the approximation of functions as well as their derivatives integrals and primitives especially in the case of equidistant interpolation nodes these infinitely smooth interpolants offer a much better choice than their polynomial analogue a natural and important question concerns the condition of this rational approximation method in this paper we extend a recent study of the lebesgue function and constant associated with berrut s rational interpolant at equidistant nodes to the family of floater hormann interpolants which includes the former as a special case

symmetrical formulas for rational interpolants abstract symmetrical determinantal formulas for the numerator and denominator of an ordinary rational interpolant are presented and discussed degenerate cases are analysed

stability of localized operators let lp 1 p be the space of all p summable sequences and ca be the convolution operator associated with a summable sequence a it is known that the lp stability of the convolution operator ca for different 1 p are equivalent to each other i e if ca has lp stability for some 1 p then ca has lq stability for all 1 q in the study of spline approximation wavelet analysis time frequency analysis and sampling there are many localized operators of non convolution type whose stability is one of the basic assumptions in this paper we consider the stability of those localized operators including infinite matrices in the sjostrand class synthesis operators with generating functions enveloped by shifts of a function in the wiener amalgam space and integral operators with kernels having certain regularity and decay at infinity we show that the lp stability or lp stability of those three classes of localized operators are equivalent to each other and we also prove that the left inverse of those localized operators are well localized

time frequency analysis of localization operators abstract we study a class of pseudodifferential operators known as time frequency localization operators anti wick operators gabor toeplitz operators or wave packets given a symbol a and two windows 1 2 we investigate the multilinear mapping from a 1 2 s r 2d s r d s r d to the localization operator aa 1 2 and we give sufficient and necessary conditions for aa 1 2 to be bounded or to belong to a schatten class our results are formulated in terms of time frequency analysis in particular we use modulation spaces as appropriate classes for symbols and windows

lattice boltzmann method for compressible flows with high mach numbers in this paper we present a lattice boltzmann model to simulate compressible flows by introducing an attractive force this scheme has two main advantages one is to soften sound speed effectively which greatly raises the mach number up to 5 another is its relative simple procedure simulations of the march cone and the comparison between theoretical expectations and simulations demonstrate that the scheme is effective in the simulation of compressible flows with high mach numbers which would create many new applications

wiener s lemma for infinite matrices in this paper we introduce a class of infinite matrices related to the beurling algebra of periodic functions and we show that it is an inverse closed subalgebra of mathcal b ell q w the algebra of all bounded linear operators on the weighted sequence space ell q w for any 1 q and any discrete muckenhoupt aq weight w

banach spaces related to integrable group representations and their atomic decompositions i abstract we present a general theory of banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states i e the atoms arise from a single element under the group action several well known decomposition theories are contained as special examples and are unified under the aspect of group theory

construction of new generating function based on linear barycentric rational interpolation for numerical solution of fractional differential equations abstract in this paper a new generating function based on linear barycentric rational interpolation is introduced for the numerical solution of fractional differential equations which is designed on fractional linear multistep methods the consistency of the proposed method is analyzed considering monotonicity of the coefficients the stability of the proposed method is presented also the stability regions are determined which show the new scheme improves the stability properties and the accuracy compared with the fractional backward difference formulae finally the extracted numerical results show the effectiveness of the proposed method

adaptive linear barycentric rational finite differences method for stiff odes abstract it is our purpose to introduce a simple multistep method based on linear barycentric rational interpolation for solving ordinary differential equations also we design an adaptive version having one free parameter which is used to improve the stability properties numerical experiments of the constructed methods on some well known stiff problems indicate efficiency and capability of the methods in solving stiff problems

gabor frames and time frequency analysis of distributions abstract this paper lays the foundation for a quantitative theory of gabor expansions f x k n c k n e 2 in x g x k in analogy to wavelet expansions of besov triebel lizorkin spaces we show that the correct class of spaces which can be characterized by the magnitude of the coefficients c k n is the class of modulation spaces to analyze the behavior of the coefficients it is necessary to invert the gabor frame operator on these spaces we show that the frame operator is invertible on modulation spaces if and only if it is invertible on l 2 and the atom g is in a suitable space of test functions a similar statement for wavelet theory is false the second part is devoted to gabor analysis on general time frequency lattices

characterization of coorbit spaces with phase space covers abstract we show that coorbit spaces can be characterized in terms of arbitrary phase space covers which are families of phase space multipliers associated with partitions of unity this generalizes previously known results for time frequency analysis to include time scale decompositions as a by product we extend the existing results for time frequency analysis to an irregular setting

polynomial interpolation lagrange versus newton we show that the lagrangian form of the interpolating polynomial can be calculated with the same number of arithmetic operations as the newtonian form

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

harmonic analysis in phase space and finite weyl heisenberg ensembles weyl heisenberg ensembles are translation invariant determinantal point processes on mathbb r 2d associated with the schrodinger representation of the heisenberg group and include as examples the ginibre ensemble and the polyanalytic ensembles which model the higher landau levels in physics we introduce finite versions of the weyl heisenberg ensembles and show that they behave analogously to the finite ginibre ensembles more specifically guided by the observation that the ginibre ensemble with n points is asymptotically close to the restriction of the infinite ginibre ensemble to the disk of area n we define finite wh ensembles as adequate finite approximations of the restriction of infinite wh ensembles to a given domain omega we provide a precise rate for the convergence of the corresponding one point intensities to the indicator function of omega as omega is dilated and the process is rescaled proportionally thermodynamic regime the construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials but rather on phase space methods second we apply our construction to study the pure finite ginibre type polyanalytic ensembles which model finite particle systems in a single landau level and are defined in terms of complex hermite polynomials on a technical level we show that finite wh ensembles provide an approximate model for finite polyanalytic ginibre ensembles and we quantify the corresponding deviation by means of this asymptotic description we derive estimates for the rate of convergence of the one point intensity of polyanalytic ginibre ensembles in the thermodynamic limit

arbitrary lagrangian eulerian unstructured finite volume lattice boltzmann method for computing two dimensional compressible inviscid flows over moving bodies the objective of this study is to develop and apply an arbitrary lagrangian eulerian unstructured finite volume lattice boltzmann method ale fvlbm for solving two dimensional compressible inviscid flows around moving bodies the two dimensional compressible form of the lb equation is considered and the resulting lb equation is formulated in the ale framework on an unstructured body fitted mesh to correctly model the body shape and properly incorporate the mesh movement due to the body motion the spatial discretization of the resulting system of equations is performed by a second order cell centered finite volume method on arbitrary quadrilateral meshes and an implicit dual time stepping method is utilized for the time integration to stabilize the numerical solution appropriate numerical dissipation terms are added to the formulation at first the shock tube problem is computed to examine the accuracy of the solution obtained by applying the proposed fvlbm for this unsteady test case which includes shock expansion wave and contact discontinuity in the flow domain then the stationary isentropic vortex is simulated on both the stationary and moving meshes to assess the implementation of the geometric conservation law in enhancing the solution accuracy of the ale fvlbm the compressible inviscid flow in the transonic regime is then computed around the stationary naca0012 airfoil in order to further study the sensitivity of the solution method to the user defined parameters now the transonic inviscid flow is simulated over the pitching or plunging naca0012 airfoil to investigate the accuracy and capability of the proposed solution method ale fvlbm for the computation of the compressible flows over moving bodies finally the pitching or plunging naca0012 airfoil near the ground in the transonic inviscid flow is simulated as a practical and challenging problem to study the ground effect on the aerodynamic characteristics of the airfoil it is indicated that the solution methodology proposed based on the finite volume lbm formulated in the arbitrary lagrangian eulerian framework ale fvlbm is capable of accurately computing the compressible inviscid flows around the moving bodies with and without the ground effect

acoustic equations of state for simple lattice boltzmann velocity sets the lattice boltzmann lb method typically uses an isothermal equation of state this is not sufficient to simulate a number of acoustic phenomena where the equation of state cannot be approximated as linear and constant however it is possible to implement variable equations of state by altering the lb equilibrium distribution for simple velocity sets with velocity components i 1 0 1 for all i these equilibria necessarily cause error terms in the momentum equation these error terms are shown to be either correctable or negligible at the cost of further weakening the compressibility for the d1q3 velocity set such an equilibrium distribution is found and shown to be unique its sound propagation properties are found for both forced and free waves with some generality beyond d1q3 finally this equilibrium distribution is applied to a nonlinear acoustics simulation where both mechanisms of nonlinearity are simulated with good results this represents an improvement on previous such simulations and proves that the compressibility of the method is still sufficiently strong even for nonlinear acoustics

the linear barycentric rational quadrature method for volterra integral equations we introduce two direct quadrature methods based on linear rational interpolation for solving general volterra integral equations of the second kind the first deduced by a direct application of linear barycentric rational quadrature given in former work is shown to converge at the same rate as the rational quadrature rule but is costly on long integration intervals the second based on a composite version of this quadrature rule loses one order of convergence but is much cheaper both require only a sample of the involved functions at equispaced nodes and yield an infinitely smooth solution of most classical examples with machine precision

time frequency partitions and characterizations of modulation spaces with localization operators abstract we study families of time frequency localization operators and derive a new characterization of modulation spaces this characterization relates the size of the localization operators to the global time frequency distribution as a by product we obtain a new proof for the existence of multi window gabor frames and extend the structure theory of gabor frames

matrices for the direct determination of the barycentric weights of rational interpolation abstract let x0 xn be n 1 interpolation points nodes and 0 n be n 1 interpolation data then every rational function r with numerator and denominator degress n interpolating these values can be written in its barycentric form completely determined by a vector u of n 1 barycentric weights uk finding u is therefore an alternative to the determination of the coefficients in the canonical form of r it is advantageous inasmuch as u contains information about unattainable points and poles in classical rational interpolation the numerator and the denominator of r are made unique up to a constant factor by restricting their respective degrees we determine here the corresponding vectors u by applying a stabilized elimination algorithm to a matrix whose kernel is the space spanned by the u s the method is of complexity o n3 in terms of the denominator degree n it seems on the other hand to be among the most stable ones

two dimensional lattice boltzmann model for compressible flows with high mach number abstract in this paper we present an improved lattice boltzmann model for compressible navier stokes system with high mach number the model is composed of three components i the discrete velocity model by m watari and m tsutahara phys rev e 67 2003 036306 ii a modified lax wendroff finite difference scheme where reasonable dissipation and dispersion are naturally included iii artificial viscosity the improved model is convenient to compromise the high accuracy and stability the included dispersion term can effectively reduce the numerical oscillation at discontinuity the added artificial viscosity helps the scheme to satisfy the von neumann stability condition shock tubes and shock reflections are used to validate the new scheme in our numerical tests the mach numbers are successfully increased up to 20 or higher the flexibility of the new model makes it suitable for tracking shock waves with high accuracy and for investigating nonlinear nonequilibrium complex systems

vector valued gabor frames associated with periodic subsets of the real line the notion of vector valued frame also called superframe was first introduced by balan in the context of multiplexing it has significant applications in mobile communication satellite communication and computer area network for vector valued gabor analysis existent literatures mostly focus on l 2 r c l instead of its subspace let a 0 and s be an a z periodic measurable set in r i e s a z s this paper addresses gabor frames in l 2 s c l with rational time frequency product they can model vector valued signals to appear periodically but intermittently and the projections of gabor frames in l 2 r c l onto l 2 s c l cannot cover all gabor frames in l 2 s c l if s r by introducing a suitable zak transform matrix we characterize completeness and frame condition of gabor systems obtain a necessary and sufficient condition on gabor duals of type i resp ii for a general gabor frame and establish a parametrization expression of gabor duals of type i resp ii all our conclusions are closely related to corresponding zak transform matrices this allows us to easily realize these conclusions by designing the corresponding matrix valued functions an example theorem is also presented to illustrate the efficiency of our method

non negativity and stability analyses of lattice boltzmann method for advection diffusion equation stability is one of the main concerns in the lattice boltzmann method lbm the objectives of this study are to investigate the linear stability of the lattice boltzmann equation with the bhatnagar gross krook collision operator lbgk for the advection diffusion equation ade and to understand the relationship between the stability of the lbgk and non negativity of the equilibrium distribution functions edfs this study conducted linear stability analysis on the lbgk whose stability depends on the lattice peclet number the courant number the single relaxation time and the flow direction the von neumann analysis was applied to delineate the stability domains by systematically varying these parameters moreover the dimensionless edfs were analyzed to identify the non negative domains of the dimensionless edfs as a result this study obtained linear stability and non negativity domains for three different lattices with linear and second order edfs it was found that the second order edfs have larger stability and non negativity domains than the linear edfs and outperform linear edfs in terms of stability and numerical dispersion furthermore the non negativity of the edfs is a sufficient condition for linear stability and becomes a necessary condition when the relaxation time is very close to 0 5 the stability and non negativity domains provide useful information to guide the selection of dimensionless parameters to obtain stable lbm solutions we use mass transport problems to demonstrate the consistency between the theoretical findings and lbm solutions

symmetry and inverse closedness of matrix algebras and functional calculus for infinite matrices we investigate the symbolic calculus for a large class of matrix algebras that are defined by the off diagonal decay of infinite matrices applications are given to the symmetry of some highly non commutative banach algebras to the analysis of twisted convolution and to the theory of localized frames

an extension of matlab to continuous functions and operators an object oriented matlab system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices about eighty matlab functions from plot and sum to svd and cond have been overloaded so that one can work with our chebfun objects using almost exactly the usual matlab syntax all functions live on 1 1 and are represented by values at sufficiently many chebyshev points for the polynomial interpolant to be accurate to close to machine precision each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation differentiation integration zerofinding or transforms applications in approximation theory and numerical analysis are explored and possible extensions for more substantial problems of scientific computing are mentioned

wiener s lemma for twisted convolution and gabor frames as a consequence ca is invertible and bounded on all p zd for 1 p oo simultaneously in this article we study several non commutative generalizations of wiener s lemma and their application to gabor theory the paper is divided into two parts the first part sections 2 and 3 is devoted to abstract harmonic analysis and extends wiener s lemma to twisted convolution the second part section 4 is devoted to the theory of gabor frames specifically to the design of dual windows with good time frequency localization in particular we solve a conjecture of janssen feichtinger and one of us 17 18 9 these two topics appear to be completely disjoint but they are not the solution of the conjectures about gabor frames is an unexpected application of methods from non commutative harmonic analysis to application oriented mathematics it turns out that the connection between twisted convolution and the heisenberg group and the theory of symmetric group algebras are precisely the tools needed to treat the problem motivated by signal analysis to be more concrete we formulate some of our main results first and will deal with the details and the technical background later