an algebraic construction of generalized coherent states for shape invariant potentials generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics we show that this generalized formalism is able to a supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system continuity of labelling resolution of unity temporal stability and action identity b reproduce results already known for shape invariant systems such as harmonic oscillator double anharmonic poschl teller and self similar potentials and c point to a formalism that provides a unified description of the different kind of coherent states for quantum systems

a general scheme for the construction of minimum uncertainty coherent states of anharmonic oscillators a mixed supersymmetric algebraic approach to the construction of the minimum uncertainty coherent states of anharmonic oscillators is presented it permits generating not only the well known coherent states of the harmonic and morse oscillators but also the so far unknown coherent states of the wei hua and generalized morse and kratzer fues oscillators the method can be applied to generate superpotentials indispensable for deriving the schrodinger equation in the supersymmetric form amenable to direct solution in the susyqm scheme

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

a simple algebraic approach to coherent states for the morse oscillator minimum uncertainty coherent states and annihilation operator coherent states for the morse oscillator are derived and shown to be equivalent they reduce in the limit of small anharmonicity constant or equivalently in the limit of large well depth to the approximate coherent states derived previously from the use of generalized displacement operator

on lagrange interpolations and kramer s sampling theorem associated with self adjoint boundary value problems abstract it is known that kramer s sampling theorem and lagrange type interpolation generalize the celebrated whittaker shannon kotel nikov sampling theorem in two different directions however no direct connection between these two directions seems to be known in this article we show that kramer s sampling theorem gives nothing more than the lagrange type interpolations provided that the kernel function associated with kramer s theorem arises from a self adjoint boundary value problem with sample eigenvalues

basic sturm liouville problems this paper is devoted to studying a q analogue of sturm liouville eigenvalue problems we formulate a self adjoint q difference operator in a hilbert space some of the properties of the eigenvalues and the eigenfunctions are discussed green s function is constructed and the problem in question is inverted into a q type fredholm integral operator with a symmetric kernel the set of eigenfunctions is shown to be a complete orthogonal set in the hilbert space examples involving basic trigonometric functions are involved

on kramer s sampling theorem associated with general strum liouville problems and lagrange interpolation kramer s sampling theorem which is a generalization of shannon s sampling theorem and its relationship with lagrange interpolation is studied kramer s sampling theorem is extended to the case where the interval a b is infinite and it is shown that any function that has a sampling expansion in the scope of kramer s theorem also has a lagrange type interpolation expansion provided that the kernel associated with kramer s theorem arises from a second order sturm liouville boundary value problem whether it is regular or singular this new technique not only reproduces many of the known sampling expansions such as those for the cosine the finite hankel and the continuous legendre transforms but also generates new ones such as those for the continuous jacobi laguerre and hermite transforms

on sampling theory and basic sturm liouville systems we investigate the sampling theory associated with basic sturm liouville eigenvalue problems we derive two sampling theorems for integral transforms whose kernels are basic functions and the integral is of jackson s type the kernel in the first theorem is a solution of a basic difference equation and in the second one it is expressed in terms of basic green s function of the basic sturm liouville systems examples involving basic sine and cosine transforms are given

an irregular sampling theorem for functions bandlimited in a generalized sense it is shown that a well known irregular sampling formula for ordinary bandlimited functions is also valid for functions bandlimited in a generalized sense except for some necessary modifications

a completeness relation for the q analogue coherent states by q integration q integration is defined for the q oscillator realization of quantum groups this is used to prove a completeness relation for the q analogue of the usual coherent states these states are overcomplete

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

orientation of particle attachment and local isotropy in diffusion limited aggregation dla we simulate 50 off lattice diffusion limited aggregation dla clusters of one million particles each the probability distribution of the angle of attachment of arriving particles with respect to the local radial direction is obtained numerically for increasing cluster size n the distribution crosses over extremely accurately to a cosine with an amplitude which decreases towards zero as a power law in n from this specific viewpoint asymptotically large dla clusters are locally isotropic

a problem of l l campbell on the equivalence of the kramer and shannon sampling theorems abstract a result concerning the equivalence of the sampling theorems of kramer and whittaker shannon kotelnikov is presented in case the associated kernels are bessel functions or jacobi functions the general problem is open a generalized mehler dirichlet formula for the jacobi functions is used to give the proof in this case for the bessel functions a generalization of poisson s integral is needed it turns out that in both cases kramer s theorem gives nothing more than shannon s theorem in the sense that each function that can be sampled by kramer s theorem can also be reconstructed by the classical shannon sampling result

completeness special functions and uncertainty principles over q linear grids we derive completeness criteria for sequences of functions of the form f x n where n is the nth zero of a suitably chosen entire function using these criteria we construct complete nonorthogonal systems of fourier bessel functions and their q analogues as well as other complete sets of q special functions we discuss connections with uncertainty principles over q linear grids and the completeness of certain sets of q bessel functions is used to prove that if a function f and its q hankel transform both vanish at the points q n n 1 0 q 1 then f must vanish on the whole q linear grid qn n

on the inversion of integral transforms associated with sturm liouville problems consider the sturm liouville boundary value problem 1 1 y q x y t2y a x b 2 2 y a cos y a sin 0 3 3 y b cos y b sin 0 where q x is continuous on a b let x t be a solution of either the initial value problem 1 and 2 or 1 and 3 in this paper we develop two techniques to invert the integral f t abf x x t dx where f x l2 a b one technique is based on the construction of some biorthogonal sequence of functions and the other is based on poisson s summation formula

an algebraic construction of generalized coherent states associated with q deformed models for primary shape invariant systems generalized coherent states for primary shape invariant potential systems quantum deformed by different models are constructed using an algebraic approach based on supersymmetric quantum mechanics we show that this generalized formalism is able a to supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system b to reproduce as particular cases results already known for shape invariant systems such as standard harmonic oscillator and p schl teller potentials as well as quantum deformed harmonic oscillator models and c point to a formalism that provides a unified description of the different kind of coherent states for quantum systems deformed or not deformed

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

sampling of vector valued transforms associated with green s matrix of dirac systems recently a sampling theorem associated with regular one dimensional dirac systems has been developed integral transforms whose kernels are solutions of the systems have been reconstructed from their values at the eigenvalues in the present article the situation is considered where the kernels are replaced by green s matrix of the problem thus a vector valued sampling theorem is established for vector valued transforms a family of examples where the systems have variable coefficients and a vector valued kramer type sampling lemma are given at the end of the paper

ladder operators for the associated laguerre functions introducing the associated laguerre functions in terms of two non negative integers we obtain simultaneously and separately realization of the laddering equations with respect to each of the integers by means of two pairs of ladder operators besides two different types of shape invariance symmetries are realized this approach leads to a derivation of shape invariance equations of third type which are realized by two simultaneous raising and lowering operators of two parameters

a quadratically convergent krawczyk like algorithm in this paper we introduce a method for computing a solution of a nonlinear system which is similar to that proposed by r krawczyk computing 4 1969 pp 187 201 our method however needs considerably less work per step starting with an interval vector we give a criterion under which the method is convergent to the solution of the system if a solution is contained in the interval vector if the starting vector contains no solution then the method will break down after a finite number of steps

an integrated approach to ladder and shift operators for the morse oscillator radial coulomb and radial oscillator potentials the morse oscillator radial coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods these problems correspond to different realizations of the so 2 1 algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems the resultant transition operators act as ladder or energy changing operators in the cases of the coulomb and harmonic oscillator potentials whereas they act as shift operators acting at constant energy in the case of the morse potential this is a consequence of the so 2 1 dynamical symmetry whereby the morse hamiltonian is expressible solely in terms of the casimir operator of the algebra an alternative algebraic approach the use of the method of supersymmetric quantum mechanics or factorization produces in each case a set of shift operators relations between the various ladder and shift operators may be identified by means of the appropriate mappings and these results can be generalized so as to relate the one dimensional morse oscillator to the radial coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions

on lagrange interpolation and kramer type sampling theorems associated with sturm liouville problems this article is devoted to a connection between kramer s sampling theorem and sampling expansions generated by lagrange interpolation it is shown that any function that has a sampling expansion in the scope of kramer s theorem also has a lagrange type interpolation expansion provided that the kernel associated with kramer s theorem arises from a second order sturm liouville boundary value problem this new approach which for a variety of regular and singular sturm liouville problems leads to associated sampling theorems recovers not only many known sampling expansions but also gives new ways to calculate the corresponding sampling functions new sampling series are included

bounded rank one perturbations in sampling theory sampling theory concerns the problem of reconstruction of functions from the knowledge of their values at some discrete set of points in this paper we derive an orthogonal sampling theory and associated lagrange interpolation formulae from a family of bounded rank one perturbations of a self adjoint operator that has only discrete spectrum of multiplicity one

new techniques for the analysis of linear interval equations abstract the basic properties of interval matrix multiplication and several well known solution algorithms for linear interval equations are abstracted by the concept of a sublinear map the new concept coupled with a systematic use of ostrowski s comparison operator in a form generalized to interval matrices is used to derive quantitative information about the result of interval gauss elimination andthe limit of various iterative schemes for the solution of linear interval equations moreover optimality results are prove for 1 the use of the midpoint inverse as a preconditioning matrix and 2 gauss seidel iteration with componentwise intersection this extends and improves results by scheu krawczyk and alefeld and herzberger

generalized robertson intelligent states and squeezing for supersymmetric and shape invariant systems an algebraic construction we obtain the robertson schrodinger uncertainty relation for shape invariant systems and construct generalized robertson intelligent states for these systems using an algebraic approach based on the supersymmetric quantum mechanics using the variances of generalized quadrature operators we study the coherency and squeezing properties evaluating their dependence on different kinds of generalizations for shape invariant systems with potential parameters related by a translation poschl teller potential and by a scaling self similar potential

on the stability of multivariate trigonometric systems abstract kadec s 1 4 theorem says that if n n z is a sequence of real numbers for which n n l 1 4 then ei n n z forms a riesz basis for l2 s favier r zalik c chui and x shi extended this result to the multivariate case but their results lead to very small stability bounds in this paper we give an optimal stability bound for the multivariate trigonometric systems moreover for the case of fourier frames in l2 d we also give the stability bounds

the structure of finitely generated shift invariant spaces in l2 rd a simple characterization is given of finitely generated subspaces of l2 rd which are invariant under translation by any multi integer and is used to give conditions under which such a space has a particularly nice generating set namely a basis and more than that a basis with desirable properties such as stability orthogonality or linear independence the last property makes sense only for local spaces i e shift invariant spaces generated by finitely many compactly supported functions and special attention is paid to such spaces as an application we prove that the approximation order provided by a given local space is already provided by the shift invariant space generated by just one function with this function constructible as a finite linear combination of the finite generating set for the whole space hence compactly supported this settles a question of some 20 years standing

sampling expansions associated with kamke problems the present paper is devoted to the derivation of sampling expansions for entire functions which are represented as integral transforms where a differential operator is acting on the kernels the situation generalizes the results obtained in sampling theory associated with boundary value problems to the case when the differential equation has the form n y lambda p y where n and p are two differential expressions of orders n and p respectively n p and lambda is the eigenvalue parameter both self adjoint and non self adjoint cases will be considered with examples in which the boundary conditions are strongly regular

numerical analysis of the non uniform sampling problem we give an overview of recent developments in the problem of reconstructing a band limited signal from nonuniform sampling from a numerical analysis view point it is shown that the appropriate design of the finite dimensional model plays a key role in the numerical solution of the nonuniform sampling problem in the one approach often proposed in the literature the finite dimensional model leads to an ill posed problem even in very simple situations the other approach that we consider leads to a well posed problem that preserves important structural properties of the original infinite dimensional problem and gives rise to efficient numerical algorithms furthermore a fast multilevel algorithm is presented that can reconstruct signals of unknown bandwidth from noisy nonuniformly spaced samples we also discuss the design of efficient regularization methods for ill conditioned reconstruction problems numerical examples from spectroscopy and exploration geophysics demonstrate the performance of the proposed methods

on interval operators obtained by splitting the lipschitz matrix abstract interval operators of a function strip obtained by splitting the given lipschitz matrix a b c are discussed using these operators we can construct a sequence of intervals which converges to a limit interval if the set of solutions of a given function strip is not empty under certain conditions the limit interval is a fixed interval of the corresponding interval operator the different fixed intervals and convergence factors for some splittings are determined

sampling theory associated with q difference equations of the sturm liouville type we will show that if the kernel k x t in the representation f t a0k x t u t dqt with u l2q 0 a is a solution of a second order q sturm liouville boundary problem then f admits a representation as a sampling formula of the form f t n 0f n w t w n t n where n is the nth eigenvalue of the associated q sturm liouville boundary problem and w t is the q wronskian of two solutions selected in a specified way

krawczyk like algorithms for the solution of systems of nonlinear equations in this paper we present interval arithmetic methods for the solution of systems of nonlinear equations these methods are based on the algorithm of r krawczyk 5 and a modification introduced in 3 starting with an interval vector containing a solution of the given system these methods converge to that solution under relatively weak conditions i e it can be said that the convergence is almost global

q ν deformation of generalized basic hypergeometric states we provide with a q deformation of the generalized hypergeometric coherent states defined such that the normalization function is given by generalized basic hypergeometric functions these states are eigenstates of suitably defined deformed lowering operators we study the domain of convergence of the corresponding normalization function on the basis of these states we investigate generalized basic hypergeometric husimi distributions and corresponding phase distributions as well as new analytic basis representations of arbitrary quantum states in bargmann and hardy spaces the quantum statistical properties of the states such as photon counting statistics and quadrature squeezing are analytically and numerically discussed in the framework of conventional quantum optics

applications of krein s theory of regular symmetric operators to sampling theory the classical kramer sampling theorem establishes general conditions that allow the reconstruction of functions by mean of orthogonal sampling formulae one major task in sampling theory is to find concrete non trivial realizations of this theorem in this paper we provide a new approach to this subject on the basis of krein s theory of representation of simple regular symmetric operators having deficiency indices 1 1 we show that the resulting sampling formulae have the form of lagrange interpolation series we also characterize the space of function reconstructible by our sampling formulae our construction allows a rigorous treatment of certain ideas proposed recently in quantum gravity

sampling theorem associated with a dirac operator and the hartley transform abstract kramer s sampling theorem indicates that under certain conditions sampling theorems associated with boundary value problems involving n th order self adjoint differential operators can be obtained in this paper we extend this result and derive a sampling theorem associated with boundary value problems involving a one dimensional system of dirac operators as a special case we obtain a sampling theorem for the hartley transform of a bandlimited function

a model for the continuous q ultraspherical polynomials an algebraic interpretation for two classes of continuous q polynomials is provided rogers continuous q hermite polynomials and continuous q ultraspherical polynomials are shown to realize respectively bases for representation spaces of the q heisenberg algebra and a q deformation of the euclidean algebra in these dimensions a generating function for the continuous q hermite polynomials and a q analog of the fourier gegenbauer expansion are naturally obtained from these models

sampling theorems associated with biorthogonal q bessel functions this paper deals with the derivation of sampling theorems associated with q biorthogonal systems we derive interpolation expansions for qhankel transforms whose kernels are the second type q bessel functions j 2 z q 0 0 q 1 we investigate the eigenvalue problem whose solutions are the q bessel functions as well as its adjoint special cases and applications involving the associated q sine function are given the results are based on the conjecture that a family of q bessel functions of the second kind is a riesz basis clues are given to support our claim

more on the q oscillator algebra and q orthogonal polynomials properties of certain q orthogonal polynomials are connected to the q oscillator algebra the wall and q laguerre polynomials are shown to arise as matrix elements of q exponentials of the generators in a representation of this algebra a realization is presented where the continuous q hermite polynomials form a basis of the representation space various identities are interpreted within this model in particular the connection formula between the continuous big q hermite and the continuous q hermite polynomials is thus obtained and two generating functions for these last polynomials are derived algebraically

calculation of energy eigenvalues for the quantum anharmonic oscillator with a polynomial potential we choose the squeezed vacuum state as a one parameter trial wavefunction to minimize the energy of an anharmonic oscillator with a polynomial perturbation potential using the optimal vacuum state obtained we generate the optimal generalized number state basis and set up the hamiltonian in the normal ordered form then truncating the basis we produce finite dimensional matrices in a simple manner and find their eigenvalues by standard methods we will apply our method to the morse potential whose exact eigenvalues are known as a check on the accuracy of the results that we have derived two models of the double well and the quartic potential are considered as examples

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

real paley wiener theorems for the koornwinder swarttouw q hankel transform abstract we derive two real paley wiener theorems in the setting of quantum calculus the first uses techniques due to tuan and zayed v k tuan a i zayed paley wiener type theorems for a class of integral transforms j math anal appl 266 1 2002 200 226 in order to describe the image of the space l q 2 0 r under koornwinder and swarttouw q hankel transform t h koornwinder r f swarttouw on q analogues of the fourier and hankel transforms trans amer math soc 333 1 1992 445 461 and contains as a special case a description of the domain of the q sampling theorem associated with the q hankel transform l d abreu a q sampling theorem related to the q hankel transform proc amer math soc 133 4 2005 1197 1203 the second characterizes the image of compactly supported q smooth functions under a rescaled version of the q hankel transform and is a q analogue of a recent result due to andersen n b andersen real paley wiener theorems for the hankel transform j fourier anal appl 12 1 2006 17 25

coherent states of gompertzian growth the origin of the gompertz function g t g 0 e b a 1 e at widely applied to fit the biological and medical data particularly growth of organisms organs and tumors is analyzed it is shown that this function is a solution of a time dependent counterpart of the schrodinger equation for the morse oscillator with anharmonicity constant equal to 1 the coherent states of the gompertzian systems which minimize the time energy uncertainty relation have been found these are eigenstates of the annihilation operator identified with the operator of growth whereas eigenstates of the creation operator represent the gompertzian states of regression the coherent formation of the specific growth patterns in the gompertzian systems appears as a result of the nonlocal long range cooperation between the microlevel the individual cell and the macrolevel the system as a whole

further results on linear interval equations abstract as a continuation of my paper new techniques for the analysis of linear interval equations linear algebra appl 58 273 325 1984 the interrelation between interval gauss elimination and interval iteration is investigated main results are a new existence theorem for interval gauss elimination in the guise of a perturbation theorem a convergence and comparison theorem for a general family of interval iteration schemes and a new method for the calculation of the hull of the solution set of linear interval equations with inverse positive coefficient matrix

oversampling and aliasing in de branges spaces arising from bessel operators abstract we show that a class of de branges spaces generated by means of generalized fourier transforms associated with perturbed bessel differential equations has the properties of oversampling and aliasing

sampling in paley wiener spaces associated with fractional integro differential operators this paper employs the theory of kramer analytic kernels established by everitt et al results in math 34 1998 310 to derive sampling expansions the kernels of the recovered transforms are mittag leffler functions of two parameters these kernels are solutions of certain classes of fractional integro differential equations the technique used here is different from the analytic approach by djrbashian harmonic analysis and boundary value problems in the complex domain birkhauser basel 1993

generalized coherent squeezed and intelligent states for exactly solvable quantum systems and the analogue of the displacement and squeezing operators using an algebraic approach we construct purely coherent and squeezed states for a set of exactly solvable quantum confined systems assuming a generalization of both the ladder operator and the displacement operator methods showing the equivalence between the two procedures we obtain the analogue of the displacement and the squeeze operators and study their basic relations and their connections with the properties of the purely coherent and squeezed states that they produce in a second approach we construct generalized squeezed coherent and intelligent states for these systems considering the successive action of the displacement and the squeeze operators in the first case and the combined action of these operators in the second case we apply the generalized approach for shape invariant systems with potential parameters related by a translation and by a scaling and show that the presence of the generalizing functionals of the potential parameters in the formalism opens a large number of possibilities in the construction of these special quantum states

sampling theorems associated with fourth and higher order self adjoint eigenvalue problems abstract consider a regular self adjoint eigenvalue problem of n th order and the associated whittaker shannon kramer sampling series the samples being taken at the eigenvalues and the sinc function being replaced by the generalized fourier coefficients of the kernel arising from the eigenvalue problem with respect to the complete set of orthogonal eigenfunctions the major aim of this paper is to show that each such sampling expansion can be written as a lagrange interpolation series provided the kernel satisfies suitable conditions two concrete fourth order eigenvalue problems are examined a regular one as an application of the general result and a singular one the problem whether the two resulting sampling series are really new or can be reduced to already known results is in part open

another addition theorem for the q exponential function an addition formula with respect to a variable and parameter is established for the basic exponential function on a q quadratic grid

on the acceleration of an interval arithmetic iteration method in the paper under consideration we describe a device for the acceleration of convergence of a monotonously enclosing iterative method for solving simultaneous systems of nonlinear equations the method is a combination of the so called interval arithmetic version of the newton single step iteration method with forming intersections which computes sequences of interval vectors and a modified newton adi iterative method under certain conditions this new method is globally convergent to the solution of the nonlinear system for some classes of nonlinear equations which originate from the discretization of partial differential equations these methods are monotonously convergent under weaker assumptions than known methods which also enclose the solution repeatedly furthermore these methods compute a sequence of approximations of the solution which is faster convergent to the solution than the lower and upper bounds of the enclosing interval vectors this paper ends with some numerical results which demons