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

on the coefficients of integrated expansions of bessel polynomials a new formula expressing explicitly the integrals of bessel polynomials of any degree and for any order in terms of the bessel polynomials themselves is proved another new explicit formula relating the bessel coefficients of an expansion for infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is also established an application of these formulae for solving ordinary differential equations with varying coefficients is discussed

connection coefficients between boas buck polynomial sets abstract in this paper a general method to express explicitly connection coefficients between two boas buck polynomial sets is presented as application we consider some generalized hypergeometric polynomials from which we derive some well known results including duplication and inversion formulas

an efficient spectral method for ordinary differential equations with rational function coefficients the authors present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients they employ expansions in terms of a large class of orthogonal polynomial families including all the classical orthogonal polynomials these families obey a simple 3 term recurrence relation for differentiation which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse the inverse is an integration operator for the family and it is simply the tridiagonal coefficient matrix for the recurrence since in these families convolution operators i e matrix representations of multiplication by a function are banded for polynomials the authors are able to obtain a banded representation for linear differential operators with rational coefficients this leads to a method of solution of initial or boundary value problems that besides having an operation count that scales linearly with the order of truncation n is computationally well conditioned among the applications considered is the use of rational maps for the resolution of sharp interior layers 19 refs 2 figs 3 tabs

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

a note on irregular discrete wavelet transforms estimates of frame bounds for wavelet frames generated by discrete sets in phase space satisfying only a certain density condition are deduced numerical examples show that these estimates which are the sharpest results of this kind known to the authors are rather poor compared to daubechies estimates see ibid vol 36 no 5 p 961 1005 1990 for the usual discrete wavelet transforms

the coefficients of differentiated expansions and derivatives of ultraspherical polynomials abstract a formula expressing the ultraspherical coefficients of the general order derivative of an infinitely differentiable function in terms of its original ultraspherical coefficients is stated in a more compact form and proved in a simpler way than the formula suggested by karageorghis and phillips in their recent report 5 formulas expressing explicitly the derivatives of ultraspherical polynomials of any degree and for any order in terms of the ultraspherical polynomials are given the special cases of chebyshev polynomials of the first and second kinds and of legendre polynomials are considered an application of how to use ultraspherical polynomials for solving ordinary and partial differential equations is described

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

homogenization of nonlinear dirichlet problems in random perforated domains abstract the present paper is devoted to study the asymptotic behavior of the solutions of a dirichlet nonlinear elliptic problem posed in a perforated domain o k e where o r n is a bounded open set and k e r n a closed set similarly to the classical paper by d cioranescu and f murat each set k e is the union of disjoint closed sets k e i with critical size but while there the sets k e i were balls periodically distributed here the main novelty is that the positions and the shapes of these sets are random with a distribution given by a preserving measure n dynamical system not necessarily ergodic as in the classical result the limit problem contains an extra term of zero order the strange term which depends on the capacity of the holes relative to the nonlinear operator and also of its distribution to prove these results we introduce an original adaptation of the two scale convergence method combined with the ergodic theory

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

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

on the connection coefficients and recurrence relations arising from expansions in series of laguerre polynomials a formula expressing the laguerre coefficients of a general order derivative of an infinitely differentiable function in terms of its original coefficients is proved and a formula expressing explicitly the derivatives of laguerre polynomials of any degree and for any order as a linear combination of suitable laguerre polynomials is deduced a formula for the laguerre coefficients of the moments of one single laguerre polynomial of certain degree is given formulae for the laguerre coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its laguerre coefficients are also obtained a simple approach in order to build and solve recursively for the connection coefficients between jacobi laguerre and hermite laguerre polynomials is described an explicit formula for these coefficients between jacobi and laguerre polynomials is given of which the ultra spherical polynomials of the first and second kinds and legendre polynomials are important special cases an analytical formula for the connection coefficients between hermite and laguerre polynomials is also obtained

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

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

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

classical orthogonal polynomials dependence on parameters most of the classical orthogonal polynomials continuous discrete and their q analogues can be considered as functions of several parameters ci a systematic study of the variation innitesimal and nite of these polynomials pn x ci with respect to the parameters ci is proposed a method to get recurrence relations for connection coecients linking r c r i pn x ci t opn x ci is given and in some situations explicit expressions are obtained this allows us to compute new integrals or sums of classical orthogonal polynomials using the digamma function a basic theorem on the zeros of ci pn x ci is also proved c 2000 elsevier science b v all rights reserved msc 33c25 42c05 33b15

construction of recurrences for the coefficients of expansions in q classical orthogonal polynomials an algorithmic approach is given to construct recurrence relations for the coefficients of the series expansion of a solution of a linear q difference equation in q classical orthogonal polynomials in the hahn s sense

on moments of classical orthogonal polynomials abstract in this work using the inversion coefficients and some connection coefficients between some polynomial sets we give explicit representations of the moments of all the orthogonal polynomials belonging to the askey wilson scheme generating functions for some of these moments are also provided

second order recurrence relation for the linearization coefficients of the classical orthogonal polynomials abstract let pk be any sequence of the classical orthogonal polynomials we give explicitly a second order recurrence relation in k for the coefficients in p i p j k i j i j c i j k p k this result is obtained by a method which is alternative to the one proposed recently by a ronveaux et al applications of the result to some named systems of classical orthogonal polynomials are given

homogenization of dirichlet parabolic problems for coefficients and open sets simultaneously variable and applications to optimal design in a previous paper we studied the homogenization of a sequence of parabolic linear dirichlet problems when the coefficients and the domains vary arbitrarily here we improve the convergence result given in this paper by showing the strong convergence in l2 every time this is applied to obtain an existence result for control problems in optimal design written in a relaxed form the control variables are the material and the shape

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

on the limit relations between classical continuous and discrete orthogonal polynomials abstract limit relations between classical continuous jacobi laguerre hermite and discrete charlier meixner kravchuk hahn orthogonal polynomials are well known and can be described by relations of type lim pn x qn x deeper information on these limiting processes can be obtained from the expansion p n x k 0 r k x n k in this paper a method for the recursive computation of coefficients rk x n is designed being the main tool the consideration of a closely related connection problem which can be solved also recurrently by using an algorithm recently developed by the authors

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

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

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 the coefficients of differentiated expansions and derivatives of jacobi polynomials a formula expressing explicitly the derivatives of jacobi polynomials of any degree and for any order in terms of the jacobi polynomials themselves is proved another explicit formula which expresses the jacobi expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original jacobi coefficients is also given the results for the special case of ultraspherical polynomials are considered the results for chebyshev polynomials of the first and second kinds and for legendre polynomials are also noted an application of how to use jacobi polynomials for solving ordinary and partial differential equations is described

asymptotic behaviour of linear dirichlet parabolic problems with variable operators depending on time in varying domains abstract we study the asymptotic behaviour of the solutions of linear parabolic dirichlet problems when the coefficients and the domains where the problems are posed vary simultaneously in the limit problem it appear the h limit of the operators and as it is usual in the homogenization of dirichlet problems a new term of order zero we also obtain a corrector result

recurrence relations for connection coefficients between two families of orthogonal polynomials we describe a simple approach in order to build recursively the connection coefficients between two families of orthogonal polynomial solutions of second and fourth order differential equations

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

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

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

optimal gabor frame bounds for separable lattices and estimates for jacobi theta functions abstract we study sharp frame bounds of gabor frames for integer redundancy with the standard gaussian window and prove that the square lattice optimizes both the lower and the upper frame bound among all rectangular lattices this proves a conjecture of floch alard berrou as reformulated by strohmer beaver the proof is based on refined log convexity concavity estimates for the jacobi theta functions 3 and 4

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

expansions in series of orthogonal hypergeometric polynomials abstract let us consider an arbitrary hypergeometric polynomial qj x and a set of orthogonal hypergeometric polynomials p n x in the domain of orthogonality here the expansion coefficients of xm and xmqj x m n0 in series of the set p n x are found in terms of the polynomials x and x characterizing the second order differential equations satisfied by the involved hypergeometric polynomials the resulting general expressions which are given in an explicit and compact form are used to produce known for checking and unknown expansions for various concrete classical orthogonal polynomials

distinguishability of countable quantum states and von neumann lattice the condition for distinguishability of a countably infinite number of pure states by a single measurement is given distinguishability is to be understood as the possibility of an unambiguous measurement for a finite number of states it is known that the necessary and sufficient condition of distinguishability is that the states are linearly independent for an infinite number of states several natural classes of distinguishability can be defined we give a necessary and sufficient condition for a system of pure states to be distinguishable it turns out that each level of distinguishability naturally corresponds to one of the generalizations of linear independence to families of infinite vectors as an important example we apply the general theory to von neumann s lattice a subsystem of coherent states which corresponds to a lattice in the classical phase space we prove that the condition for distinguishability is that the area of the fundamental region of the lattice is greater than the planck constant and also find subtle behavior on the threshold these facts reveal the measurement theoretical meaning of the planck constant and give a justification for the interpretation that it is the smallest unit of area in the phase space the cases of uncountably many states and of mixed states are also discussed

bivariate krawtchouk polynomials in this paper we present a recurrent procedure to solve an inversion problem for monic bivariate krawtchouk polynomials written in vector column form giving its solution explicitly as a by product a general connection problem between two vector column of monic bivariate krawtchouk families is also explicitly solved moreover in the non monic case and also for krawtchouk families several expansion formulas are given but for polynomials written in scalar form

a note on the chebyshev coefficients of the general order derivative of an infinitely differentiable function abstract a formula expressing the chebyshev coefficients of the general order derivative of an infinitely differentiable function in terms of its chebyshev coefficients is suggested

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

minimal recurrence relations for connection coefficients between classical orthogonal polynomials discrete case abstract we present a simple approach in order to compute recursively the connection coefficients between two families of classical discrete orthogonal polynomials charlier meixner kravchuk hahn i e the coefficients cm n in the expression p n x n m 0 c m n q m x where pn x and qm x belong to the aforementioned class of polynomials this is scv2 done by adapting a general and systematic algorithm recently developed by the authors to the discrete classical situation moreover extensions of this method allow to give new addition formulae and to estimate cm n asymptotics in limit relations between some families

results for some inversion problems for classical continuous and discrete orthogonal polynomials explicit expressions for the coefficients in the expansion of classical discrete orthogonal polynomials charlier meixner krawtchouck hahn hahn eberlein into the falling factorial basis are given the corresponding inversion problems are solved explicitly this is done by using a general algorithm recently developed by the authors which is also applied to this kind of inversion problem but relating the basis and the classical continuous orthogonal polynomials of jacobi laguerre hermite and bessel

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

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

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

representations of orthogonal polynomials abstract zeilberger s algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations and an adaption of almquist and zeilberger computers recurrence and differential equations for hyperexponential integrals further versions of this algorithm allow the computation of recurrence and differential equations from rodrigues type formulas and from generating functions in particular these algorithms can be used to compute the differential difference and recurrence equations for the classical continuous and discrete orthogonal polynomials from their hypergeometric representations and from their rodrigues representations and generating functions in recent work we used an explicit formula for the recurrence equation of families of classical continuous and discrete orthogonal polynomials in terms of the coefficients of their differential difference equations to give an algorithm to identify the polynomial system from a given recurrence equation in this article we extend these results by presenting a collection of algorithms with which any of the conversions between the differential difference equation the hypergeometric representation and the recurrence equation is possible the main technique is again to use explicit formulas for structural identities of the given polynomial systems

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

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

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

determinantal processes and completeness of random exponentials the critical case for a locally finite point set lambda subset mathbb r consider the collection of exponential functions given by mathcal e lambda e i lambda x lambda in lambda we examine the question whether mathcal e lambda spans the hilbert space l 2 pi pi when lambda is random for several point processes of interest this belongs to a certain critical case of the corresponding question for deterministic lambda about which little is known for lambda the continuum sine kernel process obtained as the bulk limit of gue eigenvalues we establish that mathcal e lambda is indeed complete almost surely we also answer an analogous question on mathbb c for the ginibre ensemble arising as weak limits of the spectra of certain non hermitian gaussian random matrices in fact we establish completeness for any rigid determinantal point process in a general setting in addition we partially answer two questions of lyons and steif about stationary determinantal processes on mathbb z d

recurrences and explicit formulae for the expansion and connection coefficients in series of bessel polynomials a formula expressing explicitly the derivatives of bessel polynomials of any degree and for any order in terms of the bessel polynomials themselves is proved another explicit formula which expresses the bessel expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original bessel coefficients is also given a formula for the bessel coefficients of the moments of one single bessel polynomial of certain degree is proved a formula for the bessel coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its bessel coefficients is also obtained application of these formulae for solving ordinary differential equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between bessel bessel polynomials is described an explicit formula for these coefficients between jacobi and bessel polynomials is given of which the ultraspherical polynomial and its consequences are important special cases two analytical formulae for the connection coefficients between laguerre bessel and hermite bessel are also developed

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

linearization and connection coefficients for hypergeometric type polynomials we consider the problem of finding closed analytical formulas for both the linearization and connection coefficients for hypergeometric type polynomials directly in terms of the corresponding differential equations we illustrate the method by producing explicit formulas for hermite polynomials

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

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

the mittag leffler reproducing kernel hilbert spaces of entire and analytic functions abstract this paper investigates the function theoretic properties of two reproducing kernel functions based on the mittag leffler function that are related through a composition both spaces provide one parameter generalizations of the traditional bargmann fock space in particular the mittag leffler space of entire functions yields many similar properties to the bargmann fock space and several results are demonstrated involving zero sets and growth rates the second generalization the mittag leffler space of the slitted plane is a reproducing kernel hilbert space rkhs of functions for which caputo fractional differentiation and multiplication by z q for q 0 are densely defined adjoints of one another

efficient spectral galerkin algorithms for direct solution of second order equations using ultraspherical polynomials it is well known that spectral methods tau galerkin collocation have a condition number of o n4 n is the number of retained modes of polynomial approximations this paper presents some efficient spectral algorithms which have a condition number of o n2 based on the ultraspherical galerkin methods for second order elliptic equations in one and two space variables the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with specially structured matrices that can be efficiently inverted the complexities of the algorithms are a small multiple of nd 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions

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

an accurate solution of the poisson equation by the chebyshev tau method abstract a new tau method is presented for the two dimensional poisson equation comparison of the results for the test problem u x y sin 4 x sin 4 y with those computed by haidvogel and zang using the matrix diagonalization method and dang vu and delcarte using the chebyshev collocation method indicates that our method would be more accurate

on the coefficients of integrated expansions of ultraspherical polynomials the problem of expressing the coefficients of an expansion of orthogonal polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is considered a formula is proved for the ultraspherical polynomials of which the chebyshev and legendre polynomials are important special cases