on holder continuity of approximate solutions to parametric equilibrium problems abstract we establish verifiable sufficient conditions for holder continuity of approximate solutions to parametric equilibrium problems when solutions may be not unique many examples are provided to illustrate the need of considering approximate solutions instead of exact solutions and the essentialness of the imposed assumptions as applications we derive this holder continuity for constrained minimization variational inequalities and fixed point problems

on the asymptotic convergence of b spline wavelets to gabor functions a family of nonorthogonal polynomial spline wavelet transforms is considered these transforms are fully reversible and can be implemented efficiently the corresponding wavelet functions have a compact support it is proven that these b spline wavelets converge to gabor functions modulated gaussian pointwise and in all l sub p norms with 1

the discrete zak transform application to time frequency analysis and synthesis of nonstationary signals algorithms to compute coefficients of the finite double sum expansion of time varying nonstationary signals and to synthesize them from a finite number of expansion coefficients are presented the algorithms are based on the computation of the discrete zak transform dzt fast algorithms to compute dzt are presented modifications to the algorithms which increase robustness are given

gaussian beam based hybrid method for quasi optical systems a hybrid method for analysis of electromagnetic propagation in passive quasi optical or unbounded wave structures is presented this method is a combination of gaussian beam tracing gbt gabor expansion and the finite difference time domain method fdtd the input to a system is first expanded into a sum of gaussian beams gbs using the gabor expansion the resultant gbs are launched inside the system and tracked traced using the gbt for structures where gbt is no longer applicable the gbs form the input of an fdtd routine the output of the fdtd can further be expanded using gabor expansion and the resultant beams can be propagated through the system

double series representation of bounded signals series representations of the form f t approximately sigma sub n infinity sup infinity sigma sub k infinity sup infinity a sub n k nu t n sub e sup 2 pi kts for bounded signals f t are studied as are conditions on the unit function nu t such that coefficients a sub n k reveal the energy content of f t in the time interval n 1 2

fresnelets new multiresolution wavelet bases for digital holography we propose a construction of new wavelet like bases that are well suited for the reconstruction and processing of optically generated fresnel holograms recorded on ccd arrays the starting point is a wavelet basis of l sub 2 to which we apply a unitary fresnel transform the transformed basis functions are shift invariant on a level by level basis but their multiresolution properties are governed by the special form that the dilation operator takes in the fresnel domain we derive a heisenberg like uncertainty relation that relates the localization of fresnelets with that of their associated wavelet basis according to this criterion the optimal functions for digital hologram processing turn out to be gabor 1948 functions bringing together two separate aspects of the holography inventor s work we give the explicit expression of orthogonal and semi orthogonal fresnelet bases corresponding to polynomial spline wavelets this special choice of fresnelets is motivated by their near optimal localization properties and their approximation characteristics we then present an efficient multiresolution fresnel transform algorithm the fresnelet transform this algorithm allows for the reconstruction backpropagation of complex scalar waves at several user defined wavelength independent resolutions furthermore when reconstructing numerical holograms the subband decomposition of the fresnelet transform naturally separates the image to reconstruct from the unwanted zero order and twin image terms this greatly facilitates their suppression we show results of experiments carried out on both synthetic simulated data sets as well as on digitally acquired holograms

linear systems in gabor time frequency space a new approach for the description and operation of linear time varying systems is presented the proposed formalism implements the superposition integral describing a general linear operation in gabor time frequency space enabling the use of its joint properties the method is based on transforming both the input and the time varying impulse response function into the gabor space the 2 and 4 d arrays corresponding to the gabor spectra of the input signal and the impulse response respectively are multiplied yielding a 2 d array corresponding to the output gabor spectrum from which the output time signal is subsequently reconstructed unlike other time varying operation methods carried out in different mixed time frequency spaces e g wigner space the suggested approach when employing the critically sampled gabor expansion generates a legal output spectrum and eliminates the need to resort to cumbersome and numerically taxing approximate synthesis techniques the proposed formalism can be implemented in the areas of time variant filtering speech scrambling and any other linear operations that vary their characterizations in time

existence of solutions of generalized vector equilibrium problems in reflexive banach spaces in this paper a generalized vector equilibrium problem with set valued maps defined on a reflexive banach space is considered by using the recession method we first give the conditions under which the solution set is non empty convex and weakly compact and then extend it to the strong generalized vector equilibrium problem this facilitates generalizing and modifying various existence theorems furthermore the topological properties of the solution set are studied and it is shown that the solution set includes some boundary points

on the holder continuity of solutions to parametric multivalued vector equilibrium problems we consider parametric multivalued vector equilibrium problems of both weak and strong types in metric linear spaces sufficient conditions for the local uniqueness and holder continuity of the solutions are established as consequences some new results for variational inequalities are derived and compared with recent papers on the subject

gabor representation of generalized functions this paper deals with gabor representation of generalized functions by this we mean that a given function is to be expanded in a series involving gabor functions that are located in the points of a given lattice in the time frequency plane we shall mainly consider lattices in which the area of the elementary cells of the lattice equals 1 in 1946 gabor used these expansions for the simultaneous analysis of signals in time and frequency cf 181 gabor stated that the above mentioned expansions exist for every reasonable signal and that the coefficients in the expansion are uniquely determined by the signal this statement is true when interpreted carefully and the aim of this paper is to find out what kind of generalized functions are sufficiently well behaved to allow a development in a gabor series we shall also consider the question of the uniqueness of the coefficients in the expansions as well as questions concerning convergence it turns out that the uniqueness questions can be handled with the aid of the main results of io in particular 2 12 and 2 13 the questions on the existence of expansions are harder to answer and require special care depending on the particular function we consider lpfunctions 1 p 2 in detail and show existence of gabor representation for tempered distribution in general

robust transient signal detection using the oversampled gabor representation the gabor representation has been investigated previously as an effective and efficient means of detecting transient signals the major problem associated with the most commonly used form of the gabor representation is that it assumes that signals have integer arrival times and modulation frequencies an assumption that if violated causes signal miss match and deterioration in detector performance the present correspondence presents the concept of oversampling the gabor representation which makes the representation more robust to certain types of signal mismatch to show this the performance of an oversampled gabor detector is compared to that of critically sampled gabor detectors gabor coefficients are calculated in the oversampled representation via the least squares method which has been shown to be superior to the more commonly used biorthogonal function method in terms of corresponding detector performance

existence theorems for generalized noncoercive equilibrium problems the quasi convex case we provide a characterization of the nonemptiness of the solution set for generalized noncoercive equilibrium problems an extension of generalized quasi variational inequalities defined in reflexive banach spaces in the quasi convex case in addition several necessary and sufficient conditions for the set of solutions to these problems to be nonempty and bounded are also given our approach is based on recession notions which proved to be very useful in the study of noncoercive minimization problems in fact we find some particular cones as estimates for the recession cone of the solution set these cones for the ones containing the latter set are proved to be sharp enough to encompass several special situations found in the literature

balian low phenomenon for subspace gabor frames in this work the balian low theorem is extended to gabor also called weyl heisenberg frames for subspaces and more particularly its relationship with the unique gabor dual property for subspace gabor frames is pointed out to achieve this goal the subspace gabor frames which have a unique gabor dual of type i resp type ii are defined and characterized in terms of the zak transform for the rational parameter case this characterization is then used to prove the balian low theorem for subspace gabor frames along the same line the same characterization is used to prove a duality theorem for the unique gabor dual property which is an analogue of the ron and shen duality theorem

reproducing pairs and gabor systems at critical density abstract we use the concept of reproducing pairs to study gabor systems at critical density first we present a generalization of the balian low theorem to the reproducing pairs setting then we prove our main result that there exists a reproducing partner for the gabor system of integer time frequency shifts of the gaussian in other words the coefficients for this gabor expansion of a square integrable function can be calculated using inner products with an unstructured family of vectors in l 2 r this solves one of the last few open questions for this system

holder continuity and upper estimates of solutions to vector quasiequilibrium problems abstract in this paper we establish the holder continuity of solution mappings to parametric vector quasiequilibrium problems in metric spaces under the case that solution mappings are set valued our main assumptions are weaker than those in the literature and the results extend and improve the recent ones furthermore as an application of holder continuity we derive upper bounds for the distance between an approximate solution and a solution set of a vector quasiequilibrium problem with fixed parameters

complete gabor transformation for signal representation properties of the gabor transformation used for image representation are discussed the properties can be expressed in matrix notation and the complete gabor coefficients can be found by multiplying the inverse of the gabor 1946 matrix and the signal vector the gabor matrix can be decomposed into the product of a sparse constant complex matrix and another sparse matrix that depends only on the window function a fast algorithm is suggested to compute the inverse of the window function matrix enabling discrete signals to be transformed into generalized nonorthogonal gabor representations efficiently a comparison is made between this method and the analytical method the relation between the window function matrix and the biorthogonal functions is demonstrated a numerical computation method for the biorthogonal functions is proposed

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

sensitivity analysis for abstract equilibrium problems we develop the general framework of sensitivity analysis for equilibrium problems in the setting of vector topological normed space our approach does not make any recourse to geometrical properties and the obtained result can be viewed as an extension and generalization of the well known results on variational inequalities in the literature even though we have worked under arbitrary constraints k with holder property that have been decisive in our treatment we have obtained in a similar spirit of domokos j math anal appl 230 1999 382 389 the best lower bound for the continuity modulus despite of the properties of the boundary of k

discrete gabor transform a feasible algorithm for implementing the gabor expansion the coefficients of which are computed by the discrete gabor transform dgt is presented for a given synthesis window and sampling pattern computing the auxiliary biorthogonal function of the dgt is nothing more than solving a linear system the dgt presented applies for both finite as well as infinite sequences by exploiting the nonuniqueness of the auxiliary biorthogonal function at oversampling an orthogonal like dgt is obtained as the discrete fourier transform dft is a discrete realization of the continuous time fourier transform similarly the dgt introduced provides a feasible vehicle to implement the useful gabor expansion

generalized vector quasi equilibrium problems with applications in this paper we consider the generalized vector quasi equilibrium problem with or without involving condensing maps and prove the existence of its solution by using known fixed point and maximal element theorems as applications of our results we derive some existence results for a solution to the vector quasi optimization problem for nondifferentiable functions and vector quasi saddle point 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

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

oversampling in the gabor scheme a method for calculating the coefficients of the gabor expansion in the case of oversampling is presented the method is based on the concept of frames and utilizes the zak transform as such the zak transform highlights the meaning and importance of frames in the context of oversampling and other aspects of signal representation by means of nonorthogonal bases

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

semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems abstract in this paper we establish sufficient conditions for the solution set of parametric multivalued vector quasiequilibrium problems to be semicontinuous all kinds of semicontinuity are considered lower semicontinuity upper semicontinuity hausdorff upper semicontinuity and closedness moreover we investigate both the weak and strong solutions of quasiequilibrium problems

computationally attractive real gabor transforms we present a gabor transform for real discrete signals and present a computationally attractive method for computing the transform for the critically sampled case we derive a biorthogonal function which is very localized in the time domain therefore truncation of this biorthogonal function allows us to compute approximate expansion coefficients with significantly reduced computational requirements further truncation does not degrade the numerical stability of the transform we present a tight upper bound on the reconstruction error incurred due to use of a truncated biorthogonal function and summarize computational savings for example the expense of transforming a length 2048 signal using length 16 blocks is reduced by a factor of 26 over similar fft based methods with at most 0 04 squared error in the reconstruction

the generalized gabor transform the generalized gabor transform for image representation is discussed for a given function f t t spl isin r the generalized gabor transform finds a set of coefficients a sub mr such that f t spl sigma sub m spl infin sup spl infin spl sigma sub r spl infin sup spl infin spl alpha sub mr g t mt exp i2 spl pi rt t the original gabor transform proposed by d gabor 1946 is the special case of t t the computation of the generalized gabor transform with biorthogonal functions is discussed the optimal biorthogonal functions are discussed a relation between a window function and its optimal biorthogonal function is presented based on the zak 1967 transform when t t is rational the finite discrete generalized gabor transform is also derived methods of computation for the biorthogonal function are discussed the relation between a window function and its optimal biorthogonal function derived for the continuous variable generalized gabor transform can be extended to the finite discrete case efficient algorithms for the optimal biorthogonal function and generalized gabor transform for the finite discrete case are proposed

on perturbations of irregular gabor frames we give a new sufficient condition under which an irregular gabor system e ip x x q forms a bessel sequence for l2 r the bessel bound just requires a mild decay on this condition then can be used to prove stability of an irregular gabor frame under a perturbation of its generating function we go on to outline how the perturbation result can be used to extend a sufficient condition of heller for irregular gabor frames with compactly supported generator to the case of a noncompactly supported generator

holder continuity of perturbed solution set for convex optimization problems abstract in this paper we first establish sufficient conditions for the local uniqueness and holder continuity of perturbed solution set for a scalar optimization problem then by using a linear scalarization method we obtain the holder continuity of two classes of perturbed solution sets for a multiobjective programming problem respectively we also give some examples to illustrate that our main results are applicable these examples are also given to illustrate that our main results are new and different from the ones in literature

oversampled gabor representation for transient signals considers the gabor representation that uses a one sided exponential window for detection and analysis of transient signals earlier results on the critically sampled case are extended to the more practically useful oversampled case for oversampling by an integer factor the authors derive an explicit analytical expression for the dual window dual frame function required for computing the gabor representation based on this expression they develop an efficient procedure for computing the gabor coefficients finally they demonstrate the performance of the method by numerical examples

optimal bases of gaussians in a hilbert space applications in mathematical signal analysis abstract arbitrary square integrable normalized functions can be expanded exactly in terms of the gaussian basis g t a where a c smaller subsets of this highly overcomplete basis can be found which are also overcomplete e g the von neumann lattice g t amn where amn are on a lattice in the complex plane approximate representations of signals using a truncated von neumann lattice of only a few gaussians are considered the error is quantified using various p norms as accuracy measures which reflect different practical needs optimization techniques are used to find optimal coefficients and to further reduce the size of the basis whilst still preserving a good degree of accuracy examples are presented

on classification of intrinsic localized modes for the discrete nonlinear schrodinger equation abstract we consider localized modes discrete breathers of the discrete nonlinear schrodinger equation i d n dt n 1 n 1 2 n n 2 n 1 n z we study the diversity of the steady state solutions of the form n t ei tvn and the intervals of the frequency of their existence the base for the analysis is provided by the anticontinuous limit negative and large enough where all the solutions can be coded by the sequences of three symbols 0 and using dynamical systems approach we show that this coding is valid for 3 4533 and the point is a point of accumulation of saddle node bifurcations also we study other bifurcations of intrinsic localized modes which take place for and give the complete table of them for the solutions with codes consisting of less than four symbols

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

the solution existence of general variational inclusion problems we propose general variational inclusion problems which are slightly different from corresponding problems considered in several recent papers in the literature and show that they are advantageous sufficient conditions for the solution existence are established as applications we derive consequences for several special cases of variational inclusion problems quasioptimization problems equilibrium problems and implicit variational inequalities and show that they improve the results of some recent existing papers

two theorems on lattice expansions it is shown that there is a tradeoff between the smoothness and decay properties of the dual functions occurring in the lattice expansion problem more precisely it is shown that if g and g are dual then 1 at least one of h sup 1 2 g and h sup 1 2 g is n in l sup 2 r and 2 at least one of hg and g is not in l sup 2 r here h is the operator 1 4 pi sup 2 d sup 2 dt sup 2 t sup 2 the first result is a generalization of a theorem first stated by r c balian 1987 the second result is new and relies heavily on the fact that when g in w sup 2 2 s with s 1 2 1 2 1 2 1 2 and g 0 than 1 g not in l sup 2 s

on informational completeness of covariant localization observables and wigner coefficients an orthogonality theorem is obtained for square integrable representations on homogeneous spaces of a locally compact group from this a series of lemmas is derived showing the informational completeness of natural covariant localization operators as well as of the generalized wigner distributions matrix elements of the group some of these results give explicit reconstruction formulas for the quantum state from its expectation values against these families of operators the results are applied to special phase space representations of the heisenberg affine and galilei groups

wigner space synthesis of discrete time periodic signals a procedure that facilitates at least squares synthesis of periodic discrete time signals from wigner quasi distributions is proposed the scheme is based on expanding the desired time sequence on a generally nonorthogonal gabor type basis whose associated biorthogonal function presumably exists the specific basis selection may crucially affect the efficiency and quality of the ensuing synthesis procedure the cited basis type constitutes a considerable generalization over the standardly used orthogonal variety thus creating previously unavailable degrees of freedom of primary significance is the acquired capability of generating time frequency basis functions that are well localized localization is a highly desirable property that can advantageously serve in various applications it is shown and numerically demonstrated that benefits of localization as well as the fact that achieving effective time frequency basis localization renders a certain degree of oversampling unavoidable

the holder continuity of solutions to generalized vector equilibrium problems in this paper by using a weaker assumption we discuss the holder continuity of solution maps for two cases of parametric generalized vector equilibrium problems under the case that the solution map is a general set valued one but not a single valued one these results extend the recent ones in the literature several examples are given for the illustration of our results