universal approximations of discrete time control systems over finite time we present an algorithm for obtaining approximate internal descriptions for a discrete time control system given a finite number of external experiments input output data the system is required only to be causal with a continuous input output map the experiments should be done in a prescribed way with bounded inputs acting over a finite period of time the approximation will be valid over the ranges of inputs and time covered by the experiments the algorithm can be considered as an exact partial realization procedure for state affine systems an e stable computer program will be sketched

on simple oversampled a d conversion in shift invariant spaces it has been found that the quantization error e for a conventional oversampled analog to digital a d conversion behaves like spl par e spl par sup 2 o spl tau sup 2 with respect to the sampling rate spl tau recently conventional a d conversion has been extended to a d conversion based on shift invariant spaces as consequences of such extension it offers rich choices to build a nonideal a d conversion system of high accuracy and low computational complexity as well as reduces the noise sensitivity and computational complexity in digital to analog d a conversion therefore it is necessary to establish the estimate of quantization error for the extended a d conversion based on shift invariant spaces in this paper we introduce a constructive method to establish an estimate of the quantization error as e sup 2 o spl tau sup 2 for oversampled a d conversion in shift invariant spaces meanwhile we demonstrate that the bit rate required to encode the converted digital signal in such a d conversion scheme only increases as the logarithm of the sampling ratio therefore the quantization error is an exponentially decaying function of the bit rate in order to establish such an estimate we need the nonuniform sampling theorem for shift invariant spaces which as the necessary preparation is studied prior to introducing the constructive method

center manifolds and contractions on a scale of banach spaces we give a new proof for the existence of a ck center manifold at a nonhyperbolic equilibrium point of a finite dimensional vector field of class ck the problem is reduced to a fixed point problem on a scale of banach spaces these banach spaces consist of mappings with a certain maximal exponential growth at infinity we give conditions under which there is a unique fixed point depending differentiably on the parameters the main difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on the scale of banach spaces

the zak transform and sampling theorems for wavelet subspaces the zak transform is used for generalizing a sampling theorem of g waiter see ieee trans informat theory vol 38 p 881 884 1992 for wavelet subspaces cardinal series based on signal samples f a n n in z with a possibly unequal to 0 waiter s case are considered the condition number of the sampling operator and worst case aliasing errors are expressed in terms of zak transforms of scaling function and wavelet this shows that the stability of the resulting interpolation formula depends critically on a

spinodal decomposition for the cahn hilliard equation in higher dimensions nonlinear dynamics this paper addresses the phenomenon of spinodal decomposition for the cahn hilliard equation where n n 1 2 3 is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u 3 based on the results of 26 the nonlinear cahn hilliard equation will be discussed this equation generates a nonlinear semiflow in certain affine subspaces of h 2 in a neighborhood u e with size proportional to e n around the constant solution where lies in the spinodal region we observe the following behavior within a local inertial manifold containing there exists a finite dimensional invariant manifold which dominates the behavior of all solutions starting with initial conditions from a small ball around with probability almost 1 the dimension of is proportional to e n and the elements of exhibit a common geometric quantity which is strongly related to a characteristic wavelength proportional to e

the singular limit dynamics of semilinear damped wave equations on considere la dynamique qualitative d une equation d onde amortie semi lineaire a une dimension dans la limite singuliere correspondant a l annulation du coefficient de la derivee d ordre 2 en temps

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

shift invariant and sampling spaces associated with the fractional fourier transform domain shift invariant spaces play an important role in sampling theory multiresolution analysis and many other areas of signal and image processing a special class of the shift invariant spaces is the class of sampling spaces in which functions are determined by their values on a discrete set of points one of the vital tools used in the study of sampling spaces is the zak transform the zak transform is also related to the poisson summation formula and a common thread between all these notions is the fourier transform in this paper we extend some of these notions to the fractional fourier transform frft domain first we introduce two definitions of the discrete fractional fourier transform and two semi discrete fractional convolutions associated with them we employ these definitions to derive necessary and sufficient conditions pertaining to frft domain under which integer shifts of a function form an orthogonal basis or a riesz basis for a shift invariant space we also introduce the fractional zak transform and derive two different versions of the poisson summation formula for the frft these extensions are used to obtain new results concerning sampling spaces to derive the reproducing kernel for the spaces of fractional band limited signals and to obtain a new simple proof of the sampling theorem for signals in that space finally we present an application of our shift invariant signal model which is linked with the problem of fractional delay filtering

minimal realization of nonlinear systems described by input output difference equations we consider a general class of nonlinear discrete time systems described by their input output i o relationship we first present the necessary and sufficient conditions for existence of a local observable state space realization of the system and an explicit algorithm for computing it in cases where the i o map and its resulting realization are nonminimal we formulate an algorithm for extracting its minimal realization whenever possible the developed theory and algorithms are illustrated by means of several examples

absolute continuity of stable foliations for systems on banach spaces abstract we prove the absolute continuity of stable foliations for c 1 maps of banach spaces satisfying a globally defined infinitesimal invariant cones condition proofs of regularity for center and stable manifolds needed for the main theorem are included our results are applicable to dynamical systems generated by ordinary partial or functional differential equations including non autonomous differential equations that are periodic in time

asymptotic error expansion of wavelet approximations of smooth functions ii we generalize earlier results concerning an asymptotic error expansion of wavelet approximations the properties of the monowavelets which are the building blocks for the error expansion are studied in more detail and connections between spline wavelets and euler and bernoulli polynomials are pointed out the expansion is used to compare the error for different wavelet families we prove that the leading terms of the expansion only depend on the multiresolution subspaces v j and not on how the complementary subspaces w j are chosen consequently for a fixed set of subspaces v j the leading terms do not depend on the fact whether the wavelets are orthogonal or not we also show that daubechies orthogonal wavelets need in general one level more than spline wavelets to obtain an approximation with a prescribed accuracy these results are illustrated with numerical examples

aliasing and oblique dual pair designs for consistent sampling abstract in this paper we study some aspects of oblique duality between finite sequences of vectors f and g lying in finite dimensional subspaces w and v respectively we compute the possible eigenvalue lists of the frame operators of oblique duals to f lying in v we compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for f with norm restrictions as an application we show that these optimal duals are the closest to being tight frames and therefore have their spectrum as concentrated as possible among oblique duals with norm restrictions we obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces v and w has in oblique duality we apply this analysis to compute those rigid rotations u for w such that the canonical oblique dual of u f minimize every convex potential we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations u for w such that the canonical oblique dual pair associated to u f minimize the aliasing we point out that these two last problems are intrinsic to oblique duality within the context of consistent sampling

inertial manifolds the non self adjoint case abstract in contrast with the existing theories of inertial manifolds which are based on the self adjoint assumption of the principal differential operator in this paper we show that for general dissipative evolutionary systems described by semilinear parabolic equations with principal differential operator being sectorial and having compact resolvent there exists an inertial manifold provided that certain gap conditions hold we also show that by using an elliptic regularization this theory can be extended to a class of kdv equations where the principal differential operator is not sectorial

modeling of nonlinear discrete time systems from input output data abstract an algorithm is presented that produces a polynomial state affine model for a discrete time nonlinear system the method uses only input output information and is based on a difference equation approximation to the input output map from the difference equation a behaviour matrix is constructed to be used in the state affine realization algorithm the difference equation approach requires the estimation of fewer parameters than the usual volterra series a state space model is then obtained directly from the difference equation as a byproduct a method for calculating the volterra series directly from the difference equation is obtained practical applications to nonlinear systems are included

perturbation of regular sampling in shift invariant spaces for frames perturbation theorems for regular sampling in shift invariant spaces are derived using a generalized perturbation theorem for frames in a hilbert space which generalizes the irregular sampling theorem established by chen using this generalized irregular sampling theorem estimates for the maximum perturbation are obtained some typical examples illustrate the result

bilinear realizability is equivalent to existence of a singular affine differential i o equation abstract for continuous time analytic input output maps existence of a singular differential equation relating derivatives of controls and outputs is shown to be equivalent to bilinear realizability a similar result holds for the problem of immersion into bilinear systems the proof is very analogous to that of the corresponding and previously known result for discrete time

discrete zak transforms polyphase transforms and applications we consider three different versions of the zak 1967 transform zt for discrete time signals namely the discrete time zt the polyphase transform and a cyclic discrete zt in particular we show that the extension of the discrete time zt to the complex z plane results in the polyphase transform an important and well known concept in multirate signal processing and filter bank theory we discuss fundamental properties relations and transform pairs of the three discrete zt versions and we summarize applications of these transforms in particular the discrete time zt and the cyclic discrete zt are important for discrete time gabor 1946 expansion weyl heisenberg frame theory since they diagonalize the weyl heisenberg frame operator for critical sampling and integer oversampling the polyphase representation plays a fundamental role in the theory of filter banks especially dft filter banks simulation results are presented to demonstrate the application of the discrete zt to the efficient calculation of dual gabor windows tight gabor windows and frame bounds

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

on the relation between continuous and discrete nonlinear parametric models abstract actual applications of system identification and modeling utilize discrete time sampled data and yield discrete time models although the actual physical processes subserving the system may occur in continuous time in the case of nonlinear parametric models the physical interpretation of the estimated model parameters may change considerably between discrete time difference equation and continuous time differential equation representations furthermore a continuous time model may be desired although only discrete time data are available thus methods for defining the equivalence between these two forms of nonlinear parametric models are critically needed to assist in model development and interpretation this paper presents the kernel invariance method which is a conceptual extension of the impulse invariance method in linear system modeling as the means for defining the equivalence between continuous discrete nonlinear models

volterra series arising from the discrete schrodinger wave equation in l2 a b space the wave solution of the stationary schrodinger wave equation has been studied extensively however after removing restriction on potential function the problem of obtaining the wave solution becomes considerably difficult methods available in the literature for the closed form wave solution are applicable to specific potential for this reason it is important to strive for the wave solution with a general class of potential functions this paper introduces the volterra series framework to address this limitation the volterra system is a natural extension of linear systems such systems take into account quadratic cubic and more generally polynomial kind of non linearities in the systems the volterra system is a non linear system with the structure of the volterra series here we decompose first a wave function in an appropriate function space and subsequently the wave equation is discretized as a result of this we obtain a bilinear system treating potential as input and wave function as output such an analysis is based on inherent bilinear character of the schrodinger equation in which potential is bilinearly coupled to the wave function this paper discusses state space methodology to recover the volterra series from the bilinear systems the volterra series is attractive from the system theoretic point of view since it describes the output of a class of non linear systems explicitly in terms of the input rather than involving coupling terms and allows substantial simplifications for the numerical simulation a simulation result is introduced to demonstrate the efficacy of the proposed analysis

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

design of optimum nyquist signals based on generalized sampling theory for data communications a new method is given for the optimal design of bandlimited nyquist type signal shapes for data communications which maximizes its energy in a given time interval the method is based on the periodically nonuniform sampling pns theory making use of the linear splines the computation is straightforward and the constraint for intersymbol interference is shown to be easy to include in the problem a numerical example is given and performance of the optimal signal shapes is compared with that resulting from the use of previously obtained signal shapes in the literature it is also concluded that the optimal signal shapes thus obtained are almost immune to small offsets at the sampling instants

generalized average sampling in shift invariant spaces average sampling is motivated by realistic needs e g physical limitation of acquisition devices or characteristics of sampling procedure as an extension of the average sampling we study generalized average sampling in shift invariant spaces with frame generators in which averages are taken from suitable channeled version of signals illustrative examples are given

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

vector sampling expansions in shift invariant subspaces multi input multi output mimo sampling scheme which is motivated by applications in multi channel deconvolution and multi source separation has been investigated in many aspects common for most of results on mimo systems is that the input signals are supposed to be band limited in this paper we study vector sampling expansions on general finitely generated shift invariant subspaces necessary and sufficient conditions for a vector sampling theorem to hold are given we also give several examples to illustrate the main result

center manifolds for infinite dimensional nonautonomous differential equations abstract we study a nonlinear integral equation for a center manifold of a semilinear nonautonomous differential equation having mild solutions we assume that the linear part of the equation admits in a very general sense a decomposition into center and hyperbolic parts the center manifold is obtained directly as the graph of a fixed point for a lyapunov perron type integral operator we prove that this integral operator can be factorized as a composition of a nonlinear substitution operator and a linear integral operator the operator is formed by the green s function for the hyperbolic part and composition operators induced by a flow on the center part we formulate the usual gap condition in spectral terms and show that this condition is in fact a condition of boundedness of on corresponding spaces of differentiable functions this gives a direct proof of the existence of a smooth global center manifold

c1approximations of inertial manifolds for dissipative nonlinear equations abstract in this paper we study a class of nonlinear dissipative partial differential equations that have inertial manifolds this means that the long time behavior is equivalent to a certain finite system of ordinary differential equations we investigate ways in which these finite systems can be approximated in thec1sense geometrically this may be interpreted as constructing manifolds in phase space that arec1close to the inertial manifold of the partial differential equation under such approximations the invariant hyperbolic sets of the global attractor persist

generalizations of srb measures to nonautonomous random and infinite dimensional systems we review some developments that are direct outgrowths of or closely related to the idea of srb measures as introduced by sinai ruelle and bowen in the 1970s these new directions of research include the emergence of strange attractors in periodically forced dynamical systems random attractors in systems defined by stochastic differential equations srb measures for infinite dimensional systems including those defined by large classes of dissipative pdes quasi static distributions for slowly varying time dependent systems and surviving distributions in leaky dynamical systems

volterra series for solving weakly non linear partial differential equations application to a dissipative burgers equation a method to solve weakly non linear partial differential equations with volterra series is presented in the context of single input systems the solution x z t is represented as the output of a z parameterized volterra system where z denotes the space variable but z could also have a different meaning or be a vector in place of deriving the kernels from purely algebraic equations as for the standard case of ordinary differential systems the problem turns into solving linear differential equations this paper introduces the method on an example a dissipative burgers equation which models the acoustic propagation and accounts for the dominant effects involved in brass musical instruments the kernels are computed analytically in the laplace domain as a new result writing the volterra expansion for periodic inputs leads to the analytic resolution of the harmonic balance method which is frequently used in acoustics furthermore the ability of the volterra system to treat other signals constitutes an improvement for the sound synthesis it allows the simulation for any regime including attacks and transients numerical simulations are presented and their validity are discussed

on the trade off between feasibility and performance in bilinear and state affine model based predictive control closed loop stabilizing model based predictive control mpc algorithms for discrete time bilinear and state affine models are presented stability of the closed loop is obtained through the use of an appropriate end point weighting and end point inequality constraint in this way the infinite horizon performance index is bounded from above by the objective function that is minimized in the mpc algorithm this paper presents an algorithm that aims at obtaining a large feasibility region by maximizing off line the region that is defined by the end point inequality constraint in order to improve the performance of the mpc algorithm the conservatism of the upper bound on the infinite horizon performance index is reduced in the on line computations

classical state space realizability of input output bilinear models this paper studies the realizability property of bilinear input output i o models in the classical state space form constraints on the parameters of the bilinear i o model are suggested that lead to realizable models the complete list of 2nd and 3rd order realizable input output bilinear models together with the corresponding state equations is given in the general case some subclasses of realizable bilinear models together with their state space realizations are presented including the diagonal bilinear model and the special subclass of the superdiagonal bilinear model

persistence of overflowing manifolds for semiflow this paper which is a sequel to a previous one 4 by the same authors is devoted to the persistence of overflowing manifolds and inflowing manifolds for a semiflow in a banach space we consider a c1 semiflow defined on a banach space x that is it is continuous on 0 x and for each t 0 t t x x is c1 and t t t x t t s x for all t s 0 and x x a typical example is the solution operator for a differential equation in 4 we proved that a compact normally hyperbolic invariant manifold m persists under small c1 perturbations in the semiflow we also showed that in a neighborhood of m there exist a center stable manifold and a center unstable manifold that intersect in the manifold m in 4 the compactness and invariance of the manifold m were important assumptions in the present paper we study the more general case where the manifold m is overflowing negatively invariant and the semiflow crosses the boundary transversally or inflowing positively invariant and the semiflow crosses the boundary transversally we do not assume that m is compact or finite dimensional also m is not necessarily an imbedded manifold but an immersed manifold as an example a local unstable manifold of an equilibrium point is an overflowing manifold in brief our main results on the overflowing manifolds may be summarized as follows the precise statements are given in section 2 we assume that the immersed manifoldm does not twist very much locally m is covered by the image under t t of a subset a positive distance away from boundary dt t contracts along the normal direction and does so more strongly than it does along the tangential direction and dt t has a certain uniform continuity in a neighborhood of m if the c1 perturbation t t of t t is sufficiently close to t t then t t has a unique c1 immersed overflowing manifold m nearm furthermore if t t isck and a spectral gap condition holds then m isck similar results for inflowing manifolds are also obtained and given in section 7

stability of nonlinear polynomial arma models and their inverse stability and invertibility of discrete time polynomial arma models are studied in an extended neighbourhood to extend the local results obtained earlier the analysis combines linear robust control and classical nonlinear system theories polynomial arma models are used as examples illustrating the theory

polynomial perceptrons and their applications to fading channel equalization and co channel interference suppression this paper investigates the behaviors of polynomial perceptrons and introduces a fractionally spaced recursive polynomial perceptron with low complexity and fast convergence rate the nonlinear mapping ability of the polynomial perceptron is analyzed it is shown that a polynomial perceptron with degree l spl ges 4 satisfies the stone weierstrass theorem and can approximate any continuous function to within a specified accuracy moreover the nonlinear mapping ability of a polynomial perceptron with degree l is similar to that of the three layer perceptron with one hidden layer for time same number of neurons in the input layer the nonlinear mapping ability of the fractionally spaced recursive polynomial perceptron is also presented applications of polynomial perceptrons for fading channel equalization and co channel interference suppression in a 16 level quadrature amplitude modulation receiver system are considered computer simulations are used to evaluate and compare the performance of polynomial perceptron pp and fractionally spaced bilinear perceptron fsblp with that of the synchronous decision feedback multilayer perceptron sdfmlp fractionally spaced decision feedback multilayer perceptron fsdfmlp and the conventional decision feedback equalizer dfe the results show that the performance of the fractionally spaced bilinear perceptron is clearly superior to that of the other structures

sampling theorem for wavelet subspaces error estimate and irregular sampling the error estimate is useful in the application of the sampling theorem for the classical shannon sampling theorem various errors are widely studied but for the sampling theorem in general wavelet subspaces only the aliasing error is studied in this paper we study three other errors truncation error amplitude error and time jitter error with the same technique a result on irregular sampling is improved

irregular sampling for spline wavelet subspaces spline wavelets spl psi sub m t are important in time frequency localization due to i spl psi sub m can be arbitrarily close to the optimal case as m is sufficiently large ii spl psi sub m has compact support and simple analytic expression which lead to effective computation although the spline wavelet subspaces are so simple walter s well known sampling theorem does not hold if the order of spline m is even moreover when irregular sampling is considered in these spaces it is hard to determine the sampling density which is a serious problem in applications in this correspondence a general sampling theorem is obtained for m spl ges 3 in the sense of iterative construction and the sampling density spl delta sub m is estimated

asymmetric multi channel sampling in shift invariant spaces abstract we develop an asymmetric multi channel sampling on a shift invariant space v with a riesz generator t in l 2 r where each channeled signal is assigned a uniform but distinct sampling rate we use fourier duality between v and l 2 0 2 to find conditions under which there is a stable asymmetric multi channel sampling formula on v

recovery of missing samples for oversampling in shift invariant spaces abstract it is well known that any finite missing samples of a band limited signal can be recovered from the remaining known samples when the signal is oversampled at a rate higher than the minimum nyquist rate we consider a similar problem for oversampled signals in shift invariant spaces with compactly supported continuous riesz generators using a multi channel oversampling we find conditions under which finitely many missing samples can be recovered

third order approximate kushner filter for a non linear dynamical system the analytical and numerical solutions of the non linear exact kushner filter are not possible since the mean and variance evolutions are infinite dimensional and require the knowledge of the higher order moment evolutions the approximate filters seem to preserve some of the qualitative characteristics of the exact filter in this paper evolutions of conditional mean and conditional covariance of the third order approximate filter for estimating the states of a non linear dynamical system especially accounting state dependent and state independent noise perturbations are derived in this analysis we make a comparison of this filter with second order gaussian filter discussed in standard textbooks on non linear filtering this paper discusses duffing filter by taking up two different non linear observation equations to demonstrate the effectiveness of the higher order filters i e third order and second order filters most notably this paper is about examining the ability of the higher order filt

sampling theorem and irregular sampling theorem for multiwavelet subspaces we generalize the multiwavelet sampling theorem by reproducing a kernel that is easy to use multiscaling functions that have orthogonality regularity short compact support symmetry and high approximation order satisfy the conditions in the sampling theorem which is not possible in the case of the scalar wavelet we consider the general cases of the uniform noninteger and irregular sampling a reconstruction from more general sets of points is necessary since the measurements may not be made at uniform points finally we establish a general irregular sampling theorem for multiwavelet subspaces and derive an estimate for the perturbations of uniform noninteger sampling in shift invariant spaces

sampling and reconstruction of signals in function spaces associated with the linear canonical transform the linear canonical transform lct has been shown to be useful and powerful in signal processing optics etc many results of this transform are already known including sampling theory most existing sampling theories of the lct consider the class of bandlimited signals however in the real world many analog signals arising in engineering applications are non bandlimited in this correspondence we propose a sampling and reconstruction strategy for a class of function spaces associated with the lct which can provide a suitable and realistic model for real applications first we introduce definitions of semi and fully discrete convolutions for the lct then we derive necessary and sufficient conditions pertaining to the lct under which integer shifts of a chirp modulated function generate a riesz basis for the function spaces by applying the results we present a more comprehensive sampling theory for the lct in the function spaces and further a sampling theorem which recovers a signal from its own samples in the function spaces is established moreover some sampling theorems for shift invariant spaces and some existing sampling theories for bandlimited signals associated with the fourier transform ft the fractional ft or the lct are noted as special cases of the derived results finally some potential applications of the derived theory are presented

a sampling theorem for shift invariant subspace a sampling theorem for regular sampling in shift invariant subspaces is established the sufficient necessary condition for which it holds is found then the theorem is modified to the shift sampling in shift invariant subspaces by using the zak transform finally some examples are presented to show the generality of the theorem

explicit construction of an inertial manifold for a reaction diffusion equation abstract an inertial manifold is constructed for the scalar reaction diffusion equation u t vu xx u with a cubic nonlinearity uniform bounds are obtained for the number of zeros along solutions to the variational equations satisfied by the difference of two elements on the unstable manifolds of equilibria this uniformity leads to the global parameterization of the attractor as a function defined in the linear unstable manifold of the least stable equilibrium by the introduction of local techniques near each equilibrium we succeed in constructing an inertial manifold of lowest possible dimension

on the aliasing error in wavelet subspaces shannon s sampling formula has been extended to subspaces v j j z in multiresolution analysis when we apply the corresponding sampling formula for v 0 to a function which does not belong to this space the so called aliasing error arises this paper deals with the aliasing error ef whenever f is in v 1 we obtain an expression for its fourier transform ef which allows us to give estimations of the aliasing error both in l 2 r norm and in l r norm finally we carry out a study of the aliasing error in the case when f belongs to v 2

estimate of aliasing error for non smooth signals prefiltered by quasi projections into shift invariant spaces an ideal analog to digital a d converter prefilters a signal by an ideal lowpass filter recent research on a d conversion based on shift invariant spaces reveals that prefiltering signals by quasiprojections into shift invariant spaces provides more flexible choices in designing an a d conversion system of high accuracy this paper focuses on the accuracy of such prefiltering in which the aliasing error e sub f sup spl lambda is found to behave like e sub f sup spl lambda sub 2 o spl lambda sup spl alpha with respect to the dilation spl lambda of the underlying shift invariant space provided that the input signal f is lipschitz spl alpha continuous a formula to calculate the coefficient of the decay rate is also figured out in this paper

a stability condition for certain bilinear systems the authors derive a simple sufficient condition for the output of a discrete time time invariant bilinear system to be bounded whenever the input signal to the system is bounded by a finite constant

smoothness of inertial manifolds the theory of invariant manifolds plays an important role in the study of dynamics of nonlinear systems in finite dimensional or infinite dimensional spaces such invariant manifolds include stable manifolds unstable manifolds and center manifolds see pliss 36 for example recently the theory of inertial manifolds has been developed for some dissipative evolution equations the inertial manifold introduced by foias sell and ttmam 141 is a finite dimensional lipschitz invariant manifold attracting solutions exponentially one of the important properties of inertial manifolds is that they contain the global attractors thus the study of dynamics of infinite dimensional nonlinear systems can be reduced to the study of dynamics of flows on the inertial manifold which in turn is described by the dynamics of an ordinary differential equation there are extensive works on this subject see for example constantin 3 constantin foias nicolaenko

interpolatory filter banks and interpolatory wavelet packets abstract in classic wavelet sampling research is generally focused on interpolatory bases in individual spaces consideration is seldom given to relations among different interpolatory bases including those potentially related by filter banks in this paper we show that when an interpolatory basis exists in a space with a multiresolution analysis this space can be decomposed orthogonally into two subspaces in which interpolatory bases also exist furthermore similarly to standard orthogonal bases these interpolatory bases can be related by pairs of filters called interpolatory filter banks for such interpolatory filter banks it is shown that existence of interpolatory scaling functions generally leads to that of interpolatory wavelets and interpolatory wavelet packets with this result we further propose new algorithms for constructing such interpolatory wavelets wavelet packets from corresponding scaling functions in examples our algorithms are applied to some typical wavelet spaces demonstrating our theorems for interpolatory wavelets interpolatory wavelet packets and interpolatory filter banks

scaling functions robust to translations the discrete wavelet transform dwt is popular in a wide variety of applications its sparse sampling eliminates redundancy in the representation of signals and leads to efficient processing however the dwt lacks translation invariance this makes it ill suited for many problems where the received signal is the superposition of arbitrarily shifted replicas of a transmitted signal as when multipath occurs for example the paper develops algorithms for the design of orthogonal and biorthogonal compact support scaling functions that are robust to translations our approach is to maintain the critical sampling of the dwt while designing multiresolution representations for which the coefficient energy redistributes itself mostly within each subband and not across the entire time scale plane we obtain expedite algorithms by decoupling the optimization from the constraints on the scaling function examples illustrate that the designed scaling function significantly improves the robustness of the representation

translation invariance and sampling theorem of wavelet the sampling theorem for wavelet spaces built by walter 1992 lacks the translation invariance except for walter s weak translation invariant wavelet i e meyer s wavelet indeed we must know a priori the shift offset a in the samples f n a n spl isin z otherwise the waveform cannot be recovered since the interpolation function is dependent on this offset in this correspondence we generalize our metric functional to metrize weak shiftability and find a somewhat surprising result that the b spline wavelets of order n spl ges 3 are degenerate shiftable thus we can recover approximately the waveform by double sampling without any information on shift offset a

a new suboptimal approach to the filtering problem for bilinear stochastic differential systems the aim of this paper is to present a new approach to the filtering problem for the class of bilinear stochastic multivariable systems consisting in searching for suboptimal state estimates instead of the conditional statistics as a first result a finite dimensional optimal linear filter for the considered class of systems is defined then the more general problem of designing polynomial finite dimensional filters is considered the equations of a finite dimensional filter are given producing a state estimate which is optimal in a class of polynomial transformations of the measurements with arbitrarily fixed degree numerical simulations show the effectiveness of the proposed filter

smooth conjugacy of centre manifolds in this paper we prove that for a system of ordinary differential equations of class c r 1 1 r 0 and two arbitrary c r 1 1 local centre manifolds of a given equilibrium point the equations when restricted to the centre manifolds are c r conjugate the same result is proved for similinear parabolic equations the method is based on the geometric theory of invariant foliations for centre stable and centre unstable manifolds

construction of inertial manifolds by elliptic regularization abstract in many cases an inertial manifold 2 m for an infinite dimensional dissipative dynamical system can be represented as the graph of a smooth function from a finite dimensional hilbert space hp to another hilbert space hq the invariance property of m means that can be written as the solution of a first order partial differential equation d p g1 p p a p g2 p p 0 over hp where g1 and g2 are nonlinear functions which depend on the original dynamical system and a is a suitably stable linear operator in this paper we use a method introduced by sacker r j sacker a new approach to the perturbation theory of invariant surface comm pure appl math 18 1965 717 732 for the study of finite dimensional dynamical systems to find inertial manifolds in the infinite dimensional setting this method involves replacing the first order equation for by the regularized elliptic equation e d p g1 p p a p g2 p p with suitable boundary conditions it is shown that if a satisfies a spectral gap condition then the solutions e of the elliptic equation converge to a weak solution of 0 as e 0 furthermore m graph is an invariant manifold for the given dynamical system

inertial manifolds for a smoluchowski equation on the unit sphere the existence of inertial manifolds for a smoluchowski equation a nonlinear fokker planck equation on the unit sphere which arises in modeling of colloidal suspensions is investigated a nonlinear and nonlocal transformation is used to eliminate the gradient from the nonlinear term

a connection between multi linear and volterra systems the volterra system is a non linear system with the structure of the volterra series the volterra series is attractive from the system theoretic point of view since it enables to obtain the output of a class of non linear systems in terms of the input explicitly rather than involving input output coupling terms and allows substantial simplifications for the numerical simulation the volterra system allows to derive the stability condition as well i e obtain a bound on the output for a given bound on the input function especially for the bilinear system the bilinear system possesses the volterra series this paper derives the volterra series formalism from the multi linear system involving the coupling term attributed to k 1 th order non linearity and output function where 1

the m band cardinal orthogonal scaling function the m band symmetric cardinal orthogonal scaling function with compact support is of interest in several applications such as sampling theory signal processing computer graphics and numerical algorithms in this paper we provide a complete mathematical analysis for the m band symmetric cardinal orthogonal scaling function firstly we generalize some results of the cardinal orthogonal scaling function from the special case m 2 to the most general case m 2 also we find some new results secondly we obtain the characterizations of the m band symmetric cardinal orthogonal scaling function and revisit some known examples to prove our theory

distance of attractors of reaction diffusion equations in thin domains abstract in this work we consider a dissipative reaction diffusion equation in a d dimensional thin domain shrinking to a one dimensional segment and obtain good rates for the convergence of the attractors to accomplish this we use estimates on the convergence of inertial manifolds as developed previously in 6 and shadowing theory

the cardinal orthogonal scaling function and sampling theorem in the wavelet subspaces in this paper we derive that there is a relation between the lowpass filter coefficient and wavelet s samples in its integer points when a scaling function is a cardinal orthogonal scaling function and we give some examples in which the lowpass filter coefficients are constructed from the wavelets then we discuss the symmetry property of cardinal orthogonal scaling function and give some useful characterizations therefore we construct a family of cardinal orthogonal scaling functions with exponential decay and higher approximation order furthermore some examples with exponential decay and higher approximation order are given and they will testify our results in the end we deduce that there no exists a cardinal orthogonal scaling function with exponential decay high approximation order and symmetry property in this family of cardinal orthogonal scaling functions constructed in our paper

generalized sampling theorems in multiresolution subspaces it is well known that under very mild conditions on the scaling function multiresolution subspaces are reproducing kernel hilbert spaces rkhss this allows for the development of a sampling theory in this paper we extend the existing sampling theory for wavelet subspaces in several directions we consider periodically nonuniform sampling sampling of a function and its derivatives oversampling multiband sampling and reconstruction from local averages all these problems are treated in a unified way using the perfect reconstruction pr filter bank theory we give conditions for stable reconstructions in each of these cases sampling theorems developed in the past do not allow the scaling function and the synthesizing function to be both compactly supported except in trivial cases this restriction no longer applies for the generalizations we study here due to the existence of fir pr banks in fact with nonuniform sampling oversampling and reconstruction from local averages we can guarantee compactly supported synthesizing functions moreover local averaging schemes have additional nice properties robustness to the input noise and compression capabilities we also show that some of the proposed methods can be used for efficient computation of inner products in multiresolution analysis after this we extend the sampling theory to random processes we require autocorrelation functions to belong to some subspace related to wavelet subspaces it turns out that we cannot recover random processes themselves unless they are bandlimited but only their power spectral density functions we consider both uniform and nonuniform sampling

sampling of periodic signals a quantitative error analysis we present an exact expression for the l sub 2 error that occurs when one approximates a periodic signal in a basis of shifted and scaled versions of a generating function this formulation is applicable to a wide variety of linear approximation schemes including wavelets splines and bandlimited signal expansions the formula takes the simple form of a parseval s like relation where the fourier coefficients of the signal are weighted against a frequency kernel that characterizes the approximation operator we use this expression to analyze the behavior of the error as the sampling step approaches zero we also experimentally verify the expression of the error in the context of the interpolation of closed curves

average sampling in shift invariant subspaces with symmetric averaging functions in this paper we study the reconstruction of functions in shift invariant subspaces from local averages with symmetric averaging functions we present an average sampling theorem for shift invariant subspaces and give quantitative results on the aliasing error and the truncation error we show that every square integrable function can be approximated by its average sampling series as special cases we also obtain new error bounds for regular sampling examples are given

on sampling in shift invariant spaces a necessary and sufficient condition for sampling in the general framework of shift invariant spaces is derived then this result is applied respectively to the regular sampling and the perturbation of regular sampling in shift invariant spaces a simple necessary and sufficient condition for regular sampling in shift invariant spaces is attained furthermore an improved estimate for the perturbation is derived for the perturbation of regular sampling in shift invariant spaces the derived estimate is easy to calculate and shown to be optimal in some shift invariant spaces the algorithm to calculate the reconstruction frame is also presented

persistence of invariant sets for dissipative evolution equations we show that results concerning the persistence of invariant sets of ordinary differential equations under perturbation may be applied directly to a certain class of partial differential equations our framework is particularly well suited to encompass numerical approximations of these partial differential equations specifically we show that for a class of pdes with ac 1 inertial form certain natural numerical approximations possess an inertial form close to that of the underlying pde in thec 1 norm

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

adaptive scheme for elimination of broadband noise and impulsive disturbances from ar and arma signals the problem of elimination of measurement noise and impulsive disturbances from autoregressive and autoregressive moving average signals is considered it is shown that the task of simultaneous detection tracking restoration can be stated as a nonlinear filtering problem and solved using the theory of extended kalman filter numerical tests carried out for real audio signals corrupted by both real and artificially generated disturbances confirm very good properties of the proposed algorithm

invariant manifolds for flows in banach spaces on presente une theorie des varietes invariantes lisses basee sur la methode classique de lyapunov penon pour des semi flots continus dans des espaces de banach

inertial manifolds associated to partly dissipative reaction diffusion systems abstract in a previous work we established the existence of finite dimensional attractors for partly dissipative reaction diffusion systems of two types systems with a polynomial growth nonlinearity and systems admitting an invariant region the aim of this article is to show that these problems possess inertial manifolds in space dimension n 2 these inertial manifolds contain the universal attractor attract exponentially all the solutions as time goes to infinity and due to the weak dissipation are here infinite dimensional our results are applied to several classical systems borrowed from mathematical biology physics and chemistry

riesz bases in l2 0 1 related to sampling in shift invariant spaces abstract the fourier duality is an elegant technique to obtain sampling formulas in paley wiener spaces in this paper it is proved that there exists an analogue of the fourier duality technique in the setting of shift invariant spaces in fact any shift invariant space v with a stable generator is the range space of a bounded one to one linear operator t between l 2 0 1 and l 2 r thus regular and irregular sampling formulas in v are obtained by transforming via t expansions in l 2 0 1 with respect to some appropriate riesz bases