sampling and interpolation in large bergman and fock spaces we obtain sampling and interpolation theorems in weighted spaces of analytic functions for radial weights of arbitrary more rapid than polynomial growth we give an application to invariant subspaces of arbitrary index in large weighted bergman spaces

adaptive computations of a posteriori finite element output bounds a comparison of the hybrid flux approach and the flux free approach two different domain decomposition approaches for the a posteriori finite element output bound method with adaptive subdomain refinement are investigated for the three dimensional convection diffusion equation the bound method provides relevant inexpensive rigorous lower and upper bounds for output linear functionals of partial differential equation pde associated with a fine mesh at a fraction of the computational cost of the fine mesh solution to achieve a desired bound gap i e difference between upper and lower output bounds at the lowest cost an adaptive subdomain refinement technique is used to refine subdomains only where needed the bound method based on a finite element domain decomposition technique requires two global calculations and a multitude of subdomain computations there are two main approaches to decompose a global domain into local subdomains for the subdomain based output bound method a hybrid flux elemental neumann subdomain approach and a flux free nodal neumann subdomain approach in this paper the two different domain decomposition approaches are briefly described and compared linear and quadratic finite element discretizations for the hybrid flux and flux free approaches are developed and validated for a convergence test and an adaptive subdomain refinement example the comparison of the hybrid flux and flux free approaches is presented in terms of numerical accuracy and computational cost

reduced order optimal control based on approximate inertial manifolds for nonlinear dynamical systems a reduced order method for optimal control problems in infinite dimensions based on approximate inertial manifolds is developed convergence of the cost optimal controls and optimal states of the finite dimensional reduced order optimal control problems to the original optimal control problem is analyzed special attention is given to the particular case when the dynamics are described by the navier stokes equations in dimension two

amplitudes on von neumann lattices a formula is derived connecting any wave function k q f in the kq representation with the corresponding amplitudes of the state f on von neumann lattices of states the formula is used for establishing a possible interpretation for these amplitudes for obtaining linear relationships between them and for finding sum rules for the squares of their absolute values and other related sum rules it can also be used for establishing completeness criteria for the lattices of states and for defining a modified hilbert space in which they become strictly complete particular attention is given to the coherent state lattice but the discussion is extended to von neumann lattices generated from an arbitrary state lattices generated from harmonic oscillator states are studied explicitly and shown incidentally to lead to a wealth of summation expressions for laguerre polynomials

phase averaged transport for quasi periodic hamiltonians for a class of discrete quasi periodic schrodinger operators defined by covariant representations of the rotation algebra a lower bound on phase averaged transport in terms of the multifractal dimensions of the density of states is proven this result is established under a diophantine condition on the incommensuration parameter the relevant class of operators is distinguished by invariance with respect to symmetry automorphisms of the rotation algebra it includes the critical harper almost mathieu operator as a by product a new solution of the frame problem associated with weyl heisenberg gabor lattices of coherent states is given

on the weyl heisenberg frames generated by simple functions let x n 0 cn e x n with cn n 0 l1 and let a 1 0 a 1 be a weyl heisenberg system e2 imx x na m n z we show that if e 0 1 and some modulo extension of e then a 1 is a frame for each 0 a 1 for certain a respectively if and only if the analytic function h z n 0 cnzn has no zero on the unit circle z z 1 these results extend the case of casazza and kalton 2002 6 that x i 1k 0 1 x ni and a 1 which brought together the frame theory and the function theory on the closed unit disk our techniques of proofs are based on the zak transform and the distribution of fractional parts of na n z

curves of every genus with many points ii asymptotically good families we resolve a 1983 question of serre by constructing curves with many points of every genus over every finite field more precisely we show that for every prime power q there is a positive constant cq with the following property for every integer g 0 there is a genus g curve over fq with at least cqg rational points over fq moreover we show that there exists a positive constant d such that for every q we can choose cq d log q we show also that there is a constant c 0 such that for every q and every n 0 and for every sufficiently large g there is a genus g curve over fq that has at least cg n rational points and whose jacobian contains a subgroup of rational points isomorphic to z nz for some r cg n

an uncertainty principle for finite frames abstract we consider the notion of uncertainty for finite frames using a difference operator inspired by the gauss hermite differential equation we obtain a time frequency measure for finite frames we then find the minimizer of the measure over all equal norm parseval frames dependent on the dimension of the space and the number of elements in the frame next we show that given a frame one can find the dual frame that minimizes this time frequency measure generalizing some work of daubechies landau and landau to the finite case and extending some recent work on sobolev duals for finite frames

interpolation and sampling hypersurfaces for the bargmann fock space in higher dimensions we study those smooth complex hypersurfaces w in open image in new window having the property that all holomorphic functions of finite weighted lp norm on w extend to entire functions with finite weighted lp norm such hypersurfaces are called interpolation hypersurfaces we also examine the dual problem of finding all sampling hypersurfaces i e smooth hypersurfaces w in open image in new window such that any entire function with finite weighted lp norm is stably determined by its restriction to w

singular values of compact pseudodifferential operators abstract this paper investigates the asymptotic decay of the singular values of compact operators arising from the weyl correspondence the motivating problem is to find sufficient conditions on a symbol which ensure that the corresponding operator has singular values with a prescribed rate of decay the problem is approached by using a gabor frame expansion of the symbol to construct an approximating finite rank operator this establishes a variety of sufficient conditions for the associated operator to be in a particular schatten class in particular an improvement of a sufficient condition of daubechies for an operator to be trace class is obtained in addition a new development and improvement of the calderon vaillancourt theorem in the context of the weyl correspondence is given additional results of this type are then obtained by interpolation

gabor super frames with hermite functions we investigate vector valued gabor frames sometimes called gabor superframes based on hermite functions hn let h h0 h1 hn be the vector of the first n 1 hermite functions we give a complete characterization of all lattices lambda subseteq mathbb r 2 such that the gabor system rm e 2 pi i lambda 2 t bf h t lambda 1 lambda lambda 1 lambda 2 in lambda is a frame for l 2 mathbb r mathbb c n 1 as a corollary we obtain sufficient conditions for a single hermite function to generate a gabor frame and a new estimate for the lower frame bound the main tools are growth estimates for the weierstrass function a new type of interpolation problem for entire functions on the bargmann fock space and structural results about vector valued gabor frames

eigenvalue estimates and mutual information for the linear time varying channel we consider linear time varying channels with additive white gaussian noise for a large class of such channels we derive rigorous estimates of the eigenvalues of the correlation matrix of the effective channel in terms of the sampled time varying transfer function and thus provide a theoretical justification for a relationship that has been frequently observed in the literature we then use this eigenvalue estimate to derive an estimate of the mutual information of the channel our approach is constructive and is based on a careful balance of the tradeoff between approximate operator diagonalization signal dimension loss and accuracy of eigenvalue estimates

estimation of the error in the reduced basis method solution of nonlinear equations the reduced basis method is a projection technique for approximating the solution curve of a finite system of nonlinear algebraic equations by the solution curve of a related system that is typically of much lower dimension in this paper the reduced basis error is shown to be dominated by an approximation error this in turn leads to error estimates for projection onto specific subspaces for example subspaces related to taylor lagrange and discrete least squares approximation

sampling theorems for shift invariant spaces gabor frames and totally positive functions we study nonuniform sampling in shift invariant spaces and the construction of gabor frames with respect to the class of totally positive functions whose fourier transform factors as hat g xi prod j 1 n 1 2 pi i delta j xi 1 e c xi 2 for delta 1 ldots delta n in mathbb r c 0 in which case g is called totally positive of gaussian type in analogy to beurling s sampling theorem for the paley wiener space of entire functions we prove that every separated set with lower beurling density 1 is a sampling set for the shift invariant space generated by such a g in view of the known necessary density conditions this result is optimal and validates the heuristic reasonings in the engineering literature using a subtle connection between sampling in shift invariant spaces and the theory of gabor frames we show that the set of phase space shifts of g with respect to a rectangular lattice alpha mathbb z times beta mathbb z forms a frame if and only if alpha beta 1 this solves an open problem going back to daubechies in 1990 for the class of totally positive functions of gaussian type the proof strategy involves the connection between sampling in shift invariant spaces and gabor frames a new characterization of sampling sets without inequalities in the style of beurling new properties of totally positive functions and the interplay between zero sets of functions in a shift invariant space and functions in the bargmann fock space

analysis of a space time continuous galerkin method for convection dominated sobolev equations abstract the convergence of space time continuous galerkin stcg method for the sobolev equations with convection dominated terms is studied in this article it allows variable time steps and the change of the spatial mesh from one time interval to the next which can make this method suitable for numerical simulations on unstructured grids we prove the existence and uniqueness of the approximate solution and get the optimal convergence rates in l h 1 norm which do not require any restriction assumptions on the space and time mesh size finally some numerical examples are designed to validate the high efficiency of the method showed herein and to confirm the correctness of the theoretical analysis

discretized gabor frames of totally positive functions in this paper a large class of universal windows for gabor frames weyl heisenberg frames is constructed these windows have the fundamental property that every overcritical rectangular lattice generates a gabor frame likewise every undercritical rectangular lattice generates a riesz sequence

the growth of bargmann functions and the completeness of sequences of coherent states the growth of bargmann functions is intimately connected to the density of the zeros of these functions and to the completeness of sequences of coherent states using these ideas we find the least density that a sequence of coherent states must have in order to be overcomplete within the space of bargmann functions of an order not exceeding and of a type not exceeding if of order these results generalize known results on the completeness of von neumann lattices the practical significance of this formalism in the context of quantum optics is also discussed

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

the zak transform and some counterexamples in time frequency analysis it is shown how the zak transform can be used to find nontrivial examples of functions f g in l sup 2 r with f g identical to 0 identical to f g where f g are the fourier transforms of f g respectively this is then used to exhibit a nontrivial pair of functions h k in l sup 2 r h not k such that mod h mod mod k mod mod h mod mod k mod a similar construction is used to find an abundance of nontrivial pairs of functions h k in l sup 2 r h not k with mod a sub h mod mod a sub k mod or with mod w sub h mod mod w sub k mod where a sub h a sub k and w sub h w sub k are the ambiguity functions and wigner distributions of h k respectively one of the examples of a pair of h k in l sup 2 r h not k with mod a sub h mod mod a sub k mod is f a grunbaum s 1981 example in addition nontrivial examples of functions g and signals f sub 1 not f sub 2 such that f sub 1 and f sub 2 have the same spectrogram when using g as window have been found

model space results for the gabor and wavelet transforms we prove that the unique gabor window leading to an analytic model space is the gaussian one and we give an analogous result for the wavelet transform for the general case we give a new approach to study irregular gabor and wavelet frames we improve some known results for gabor windows in the feichtinger algebra and for a special class of wavelets

output bounds for reduced order approximations of elliptic partial differential equations abstract we present an a posteriori finite element procedure that provides inexpensive rigorous accurate and constant free lower and upper bounds for the error in the outputs engineering quantities of interest predicted by lagrangian reduced order approximations to coercive elliptic partial differential equations the bound calculation requires i the reduced order approximation for the primal and dual field variables ii a lower bound for the minimum eigenvalue of the symmetric part of the operator and iii the solution of purely local symmetric neumann subproblems defined on small decoupled nodal overlapping patches there are two critical components to the neumann subproblems a partition of unity attenuated local residual which eliminates hybrid fluxes from both the construction and analysis of the resulting estimators while simultaneously preserving the global bound property and a l 2 regularization term which provides stability despite the absence of local nodal equilibrium of the reduced order primal and dual solutions the estimator bounding property and optimal convergence rate as the reduced order basis is enriched are proven and corroborating numerical results are presented for two examples a heat conduction fin symmetric problem and a conjugate advection diffusion multi material heat transfer non symmetric problem

gabor schauder bases and the balian low theorem the balian low theorem is a strong form of the uncertainty principle for gabor systems that form orthonormal or riesz bases for l2 r in this paper we investigate the balian low theorem in the setting of schauder bases we prove that new weak versions of the balian low theorem hold for gabor schauder bases but we constructively demonstrate that several variants of the blt can fail for gabor schauder bases that are not riesz bases we characterize a class of gabor schauder bases in terms of the zak transform and product a2 weights the riesz bases correspond to the special case of weights that are bounded away from zero and infinity

properties of a discretized coherent state representation and the relation to gabor analysis properties of a discretized coherent state representation dcsr and its connection to gabor frame analysis are discussed the dcsr approach was recently shown andersson l m 2001 j chem phys 115 1158 to yield a practical computational scheme for quantum dynamics and an iterative scheme for finding the identity operator was proposed in the present work we suggest a proof of fast convergence of the iterative scheme for computing the canonical dual to any given countable frame in a hilbert space the method of frames is concerned with the use of a non orthogonal over complete set of functions for expansion of an arbitrary function we also introduce the concept of representations of the identity operator and show how to expand arbitrary vectors using the frame elements without explicit diagonalization to an orthonormal basis numerical examples that illustrate the method are shown

the bargmann analytic representation in signal analysis the bargmann analytic representation of quantum mechanics is used in the context of time frequency signal analysis transformations between the various representations are derived examples that show how this method can be implemented numerically and demonstrate its robustness to noise are discussed

a posteriori error estimation and adaptive strategy for pgd model reduction applied to parametrized linear parabolic problems abstract we define an a posteriori verification procedure that enables to control and certify pgd based model reduction techniques applied to parametrized linear elliptic or parabolic problems using the concept of constitutive relation error it provides guaranteed and fully computable global goal oriented error estimates taking both discretization and pgd truncation errors into account splitting the error sources it also leads to a natural greedy adaptive strategy which can be driven in order to optimize the accuracy of pgd approximations the focus of the paper is on two technical points i construction of equilibrated fields required to compute guaranteed error bounds ii error splitting and adaptive process when performing pgd based model reduction performances of the proposed verification and adaptation tools are shown on several multi parameter mechanical problems

note on b splines wavelet scaling functions and gabor frames let g be a continuous compactly supported function on such that the integer translates of g constitute a partition of unity we show that the gabor system g a b with window g and time shift and frequency shift parameters a b 0 has no lower frame bound larger than 0 if b 2 3 and a 0 in particular g a b is not a gabor frame if g is a continuous compactly supported wavelet scaling function and if b 2 3 and a 0 we give an example for our result for the case that g b sub 1 the triangle function supported by 1 1 by showing pictures of the canonical dual corresponding to g a b where ab 1 4 and b crosses the lines n 2 3

pseudodifferential operators on locally compact abelian groups and sjoestrand s symbol class we investigate pseudodifferential operators on arbitrary locally compact abelian groups as symbol classes for the kohn nirenberg calculus we introduce a version of sjoestrand s class pseudodifferential operators with such symbols form a banach algebra that is closed under inversion since hard analysis techniques are not available on locally compact abelian groups a new time frequency approach is used with the emphasis on modulation spaces gabor frames and banach algebras of matrices sjoestrand s original results are thus understood as a phenomenon of abstract harmonic analysis rather than hard analysis and are proved in their natural context and generality

summability properties of gabor expansions abstract we show that there exist complete and minimal systems of time frequency shifts of gaussians in l 2 r which are not strong markushevich basis do not admit the spectral synthesis in particular it implies that there is no linear summation method for general gaussian gabor expansions on the other hand we prove that the spectral synthesis for such gabor systems holds up to one dimensional defect

on zero sets in fock spaces abstract we prove that zero sets for distinct fock spaces are not the same this is an answer of a question asked by k zhu in 6 p 209

weighted irregular gabor tight frames and dual systems using windows in the schwartz class abstract we give a characterization for the weighted irregular gabor tight frames or dual systems in l 2 r n in terms of the distributional symplectic fourier transform of a positive borel measure on r 2 n naturally associated with the system and the short time fourier transform of the windows in the case where the window or at least one of the windows in the case of dual systems belongs to s r n this result implies that for certain classes of windows such as generalized gaussians or extreme value windows the only weighted irregular gabor tight frames or even dual systems with both windows in the same class that can be constructed with these windows are the trivial ones corresponding to the measure 1 on r 2 n furthermore we show that if a such gabor system admits a dual which is of gabor type then the beurling density of the associated measure exists and is equal to one

bargmann transform zak transform and coherent states it is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the bargmann representation or by using the kq representation which was introduced by j zak in this paper both methods are considered in particular in connection with expansions of generalized functions in what are called gabor series the setting consists of two spaces of generalized functions tempered distributions and elements of the class s which appear in a natural way in the context of the bargmann transform also a thorough mathematical investigation of the zak transform is given this paper contains many comments and complements on existing literature in particular connections with the theory of interpolation of entire functions over the gaussian integers are given

density theorems for sampling and interpolation in the bargmann fock space we give a complete description of sampling and interpolation in the bargmann fock space based on a density concept of beurling roughly speaking a discrete set is a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von neumann lattice and similarly a discrete set is a set of interpolation if and only if its density in every part of the plane is strictly smaller than that of the von neumann lattice

deformation of gabor systems abstract we introduce a new notion for the deformation of gabor systems such deformations are in general nonlinear and in particular include the standard jitter error and linear deformations of phase space with this new notion we prove a strong deformation result for gabor frames and gabor riesz sequences that covers the known perturbation and deformation results our proof of the deformation theorem requires a new characterization of gabor frames and gabor riesz sequences it is in the style of beurling s characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of gabor frames without inequalities from lattices to non uniform sets

polynomials having leading terms over c2 in the fock space we characterize polynomials p having leading terms over c2 in a generalized sense as applications we study a quasi invariant subspace p generated by p in the fock space and show that a quasi invariant subspace m is similar to p if and only if m q for some polynomial q having the same leading term as p these are generalizations of guo and hou s work

coherent states for transparent potentials the darboux transformation operator method is applied to the investigation of coherent states of transparent multisoliton potentials an isometric correspondence between hilbert spaces of the states of a free particle and a particle moving in the soliton potential is established it is shown that the darboux transformation operator being unbounded and closed cannot realize an isometric mapping between hilbert spaces a quasispectral representation of transformation operators in terms of continuous basis sets is obtained different families of coherent states of the multisoliton potential are introduced measures that realize the resolution of the unity in terms of the projectors on the coherent states vectors are calculated these measures are defined by ordinary smooth functions for the states obtained with the help of bounded transformation operators and by generalized ones otherwise

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

bertini theorems over finite fields let x be a smooth quasiprojective subscheme of pn of dimension m i 0 over fq then there exist homogeneous polynomials f over fq for which the intersection of x and the hypersurface f 0 is smooth in fact the set of such f has a positive density equal to aex m 1 1 where aex s zx q s is the zeta function of x an analogue for regular quasiprojective schemes over z is proved assuming the abc conjecture and another conjecture

sampling in weighted lp spaces of entire functions in cn and estimates of the bergman kernel the necessary density condition in c known for sampling and interpolation in the lp space of entire functions with a subharmonic weight is extended to the case of a 2 homogeneous plurisubharmonic weight function in c the method is by estimating the eigenvalues of a certain toeplitz concentration operator using asymptotic estimates for the bergman kernel of independent interest

electron states in a magnetic field and random impurity potential use of the theory of entire functions the authors determine the wavefunction of an electron in the presence of a transverse magnetic field and randomly located delta function impurities in the x y plane within the subspace of the lowest landau level the wavefunction contains as a factor an entire function of z x iy which vanishes at all impurity sites the question of whether such states are extended is related to the rate of growth of an entire function in terms of the distribution of its zeros for a homogeneous distribution of impurities the corresponding weierstrass product is of order 2 and of finite type this rate of growth can be exactly compensated by the gaussian term due to the presence of the magnetic field such that there is a critical field beyond which extended states may exist if the impurities are located on the sites of a square lattice the extended states are given in closed form in terms of the weierstrass sigma function

gabor orthonormal bases generated by the unit cubes abstract we consider the problem in determining the countable sets in the time frequency plane such that the gabor system generated by the time frequency shifts of the window 0 1 d associated with forms a gabor orthonormal basis for l 2 r d we show that if this is the case the translates by elements of the unit cube in r 2 d must tile the time frequency space r 2 d by studying the possible structure of such tiling sets we completely classify all such admissible sets of time frequency shifts when d 1 2 moreover an inductive procedure for constructing such sets in dimension d 3 is also given an interesting and surprising consequence of our results is the existence for d 2 of discrete sets with g 0 1 d forming a gabor orthonormal basis but with the associated time translates of the window 0 1 d having significant overlaps

geometric conditions for multiple sampling and interpolation in the fock space we study multiple sampling interpolation and uniqueness for the classical fock spaces in the case of unbounded multiplicities we show that there are no sequences which are simultaneously sampling and interpolating when the multiplicities tend to infinity

on the invertibility of rectangular bi infinite matrices and applications in time frequency analysis abstract finite dimensional matrices with more columns than rows have no left inverses while those with more rows than columns have no right inverses we give generalizations of these simple facts to bi infinite matrices our results are then used to obtain density results for p frames of time frequency molecules in modulation spaces and identifiability results for operators with bandlimited kohn nirenberg symbols

dilation of the weyl symbol and balian low theorem the key result of this paper describes the fact that for an important class of pseudodifferential operators the property of invertibility is preserved under minor dilations of their weyl symbols this observation has two implications in time frequency analysis first it implies the stability of general gabor frames under small dilations of the time frequency set previously known only for the case that the time frequency set is a lattice secondly it allows us to derive a new balian low theorem blt for gabor systems with window in the standard window class and with general time frequency families in contrast to the classical versions of blt the new blt does not only exclude orthonormal bases and riesz bases at critical density but indeed it even excludes irregular gabor frames at critical density

an inequality for fourier laplace transforms of entire functions and the existence of exponential frames in fock space abstract we study mapping properties of the fourier laplace transform between certain spaces of entire functions we introduce a variant of the classical fock space by integrating against the monge ampere measure of the weight function and show that the norm of the fourier laplace transform in a dual fock space dominates the norm of the function equality holds when the weight function is an hermitean form as an application we get a criterium for the existence of frames of exponential functions in fock space

a tower of artin schreier extensions of function fields attaining the drinfeld vladut bound for an algebraic function fieldf having a finite constant field letg f resp n f denote the genus off resp the number of places off of degree one we construct a tower of function fields f 1 subseteq f 2 subseteq f 3 subseteq ldots over mathbb f q 2 such that the ration fi g fi tends to the drinfeld vladut boundq 1

measurement of time variant linear channels the goal of channel measurement or operator identification is to obtain complete knowledge of a channel operator by observing the image of a finite number of input signals in this paper it is shown that the spreading support of the operator that is the support of the symplectic fourier transform of the kohn nirenberg symbol of the operator has area less than one then the operator is identifiable if the spreading support is larger than one then the operator is not identifiable the shape of the support region is essentially arbitrary thereby proving a conjecture of bello the input signal considered is a weighted delta train where the weights are the window function of a finite gabor system whose elements satisfy a certain robust completeness property

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

frames in the bargmann space of entire functions we look at the decomposition of arbitrary f in l2 r in terms of the family of functions cp x w exp iimnab iman i x nb 2 with a b 0 we derive bounds and explicit formulas for the minimal expansion coefficients in the case where ab 2w n n an integer 2 2 transported to the hilbert space f of entire functions introduced by v bargmann these results are expressed as inequalities of the form

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

efficient computation of sensitivities for ordinary differential equation boundary value problems for models described by ordinary differential equation boundary value problems ode bvps we derive adjoint equations for sensitivity analysis giving explicit forms for the boundary conditions of the adjoint boundary value problem the solutions of the adjoint equations are used to efficiently compute gradients of both integral form and pointwise constraints existence and stability results are given for the adjoint system and its numerical solution the use of the method is demonstrated for a simple example where it is seen that the method is particularly advantageous for problems with more than a few parameters

beurling type density theorems in the unit disk we consider two equivalent density concepts for the unit disk that provide a complete description of sampling and interpolation ina n the banach space of functionsf analytic in the unit disk with 1 z 2 n f z bounded this study reveals a nyquist density a sequence of points is roughly speaking a set of sampling if and only if its density in every part of the disk is strictly larger thann and it is a set of interpolation if and only if its density in every part of the disk is strictly smaller thann similar density theorems are also obtained for weighted bergman spaces

a posteriori finite element output bounds in three space dimensions using the feti method abstract a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solution to the convection diffusion equation is presented the novelty is to utilize a domain decomposition method with lagrange multipliers and an iterative interface solution scheme the finite element tearing and interconnecting feti procedure to extend the bound method to three space dimensions numerical results show that the bounds are sharp their calculation consumes only a fraction of the memory required by a sparse solver and cpu usage is optimal i e proportional to the total number of d o f

curves of every genus with many points i abelian and toric families abstract let nq g denote the maximal number of f q rational points on any curve of genus g over f q ihara for square q and serre for general q proved that lim supg nq g g 0 for any fixed q here we prove limg nq g more precisely we use abelian covers of p 1 to prove lim infg nq g g log g 0 and we use curves on toric surfaces to prove lim infg nq g g1 3 0 we also show that these results are the best possible that can be proved using these families of curves

strict density inequalities for sampling and interpolation in weighted spaces of holomorphic functions abstract answering a question of lindholm we prove strict density inequalities for sampling and interpolation in fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight in particular these spaces do not admit a set that is simultaneously sampling and interpolating to prove optimality of the density conditions we construct sampling sets with a density arbitrarily close to the critical density the techniques combine methods from several complex variables estimates for and the theory of localized frames in general reproducing kernel hilbert spaces with no analyticity assumed the abstract results on fekete points and deformation of frames may be of independent interest

rational von neumann lattices a rational von neumann lattice is defined as a lattice in phase space with the constants a and b in the x and p directions given by a ratio of integers zeros of harmonic oscillator functions in the kq representation on such lattices are found it is shown that the number of zeros of the kq function determines the number of states by which a set on a von neumann lattice is overcomplete interesting relations between theta functions are derived on the basis of their connection with the harmonic oscillator states in the kq representation

a hierarchical duality approach to bounds for the outputs of partial differential equations abstract we present a technique for generating lower and upper bounds to outputs which are linear functionals of the solutions to finiteelement discretizations of symmetric or nonsymmetric coercive linear partial differential equations the method is based upon the construction of an augmented lagrangian which integrates i a quadratic energy reformulation of the desired output as the objective to be minimized with ii the finite element equilibrium equations and conforming hybridized intersubdomain continuity conditions as the constraints to be satisfied the bounds are then derived by appealing to the associated dual unconstrained max min problem evaluated for optimally chosen candidate lagrange multipliers generated by a less expensive approximation such as a low dimensional finite element discretization as in many a posteriori error estimation techniques the bound calculation requires only the solution of subdomain local symmetric neumann problems on the refined truth mesh the technique is presented and illustrated for the case of the one dimensional convection diffusion equation

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985