reproducing formulas for generalized translation invariant systems on locally compact abelian groups in this paper we connect the well established discrete frame theory of generalized shift invariant systems to a continuous frame theory to do so we let j j 2 j be a countable family of closed co compact subgroups of a sec ond countable locally compact abelian group g and study systems of the form j2jfgj p g 2 j p2pj with generators gj p in l 2 g and with each pj being a countable or an uncountable index set we refer to systems of this form as generalized translation invariant gti systems many of the familiar transforms e g the wavelet shearlet and gabor transform both their discrete and continu ous variants are gti systems under a technical local integrability condition lic we characterize when gti systems constitute tight and dual frames that yield reproducing formulas for l 2 g this generalizes results on generalized shift invariant systems where each pj is assumed to be countable and each j is a uni form lattice in g to the case of uncountably many generators and not necessarily discrete closed co compact subgroups furthermore even in the case of uniform lattices j our characterizations improve known results since the class of gti systems satisfying the lic is strictly larger than the class of gti systems sat isfying the previously used local integrability condition as an application of our characterization results we obtain new characterizations of translation invariant continuous frames and gabor frames for l 2 g in addition we will see that the admissibility conditions for the continuous and discrete wavelet and gabor trans form in l 2 r n are special cases of the same general characterizing equations

marcinkiewicz zygmund inequalities and interpolation by spherical harmonics abstract we find necessary density conditions for marcinkiewicz zygmund inequalities and interpolation for spaces of spherical harmonics in s d with respect to the l p norm moreover we prove that there are no complete interpolation families for p 2

regularization theory and neural networks architectures we had previously shown that regularization principles lead to approximation schemes that are equivalent to networks with one layer of hidden units called regularization networks in particular s

observer design for nonlinear systems with discrete time measurements this paper focuses on the development of asymptotic observers for nonlinear discrete time systems it is argued that instead of trying to imitate the linear observer theory the problem of constructing a nonlinear observer can be more fruitfully studied in the context of solving simultaneous nonlinear equations in particular it is shown that the discrete newton method properly interpreted yields an asymptotic observer for a large class of discrete time systems while the continuous newton method may be employed to obtain a global observer furthermore it is analyzed how the use of broyden s method in the observer structure affects the observer s performance and its computational complexity an example illustrates some aspects of the proposed methods moreover it serves to show that these methods apply equally well to discrete time systems and to continuous time systems with sampled outputs

landau s necessary density conditions for lca groups we derive necessary conditions for sampling and interpolation of bandlimited functions on a locally compact abelian group in line with the classical results of h landau for bandlimited functions on rd our conditions are phrased as comparison principles involving a certain canonical lattice

a simple lemma on greedy approximation in hilbert space and convergence rates for projection pursuit regression and neural network training a general convergence criterion for certain iterative sequences in hilbert space is presented for an important subclass of these sequences estimates of the rate of convergence are given under very mild assumptions these results establish an 0 1 f4n nonsampling convergence rate for projection pursuit regression and neural network training where n represents the number of ridge functions neurons or coefficients in a greedy basis expansion

nonlinear black box models in system identification mathematical foundations we discuss several aspects of the mathematical foundations of the nonlinear black box identification problem we shall see that the quality of the identification procedure is always a result of a certain trade off between the expressive power of the model we try to identify the larger the number of parameters used to describe the model the more flexible is the approximation and the stochastic error which is proportional to the number of parameters a consequence of this trade off is the simple fact that a good approximation technique can be the basis of a good identification algorithm from this point of view we consider different approximation methods and pay special attention to spatially adaptive approximants we introduce wavelet and neuron approximations and show that they are spatially adaptive then we apply the acquired approximation experience to estimation problems finally we consider some implications of these theoretical developments for the practically implemented versions of the spatially adaptive algorithms

neural approximators for nonlinear finite memory state estimation no general analytical tools are available to estimate the state of a nonlinear stochastic system observed through a nonlinear noisy channel this problem is addressed in this paper under the assumption that the statistics of the random variables are unknown hence a statistical approach is followed instead of a probabilistic one the following approximations are enforced i the state estimator is a finite memory one ii the estimation functions are given fixed structures in which a certain number of parameters have to be optimized multilayer feedforward neural networks are chosen from among various possible nonlinear approximators iii the algorithms for optimizing the parameters i e the network weights rely on a stochastic approximation simulation results are reported to compare the behaviour of the proposed estimator with the extended kalman filter and the estimators based on the on line minimization of the estimation error

landauʼs necessary density conditions for the hankel transform abstract we will prove an analogue of landau s necessary conditions h j landau necessary density conditions for sampling and interpolation of certain entire functions acta math 117 1967 37 52 for spaces of functions whose hankel transform is supported in a measurable subset s of the positive semi axis as a special case necessary density conditions for the existence of fourier bessel frames are obtained

observability of systems under uncertainty the evolution of the state x cdot of a system under uncertainty governed by a differential inclusion text for almost all t in 0 t qquad x t in f t x t is observed through an observation map h forall t in 0 t quad y t in h x t the set valued character due to the uncertainty leads to the introduction of the following sharp input output map which is the usual product forall x 0 in x quad i x 0 h circ mathcal s x 0 mathop cup limits x cdot in mathcal s x 0 h x cdot hazy input output map which is the square product forall x 0 in x quad i x 0 h square mathcal s x 0 mathop cap limits x cdot in mathcal s x 0 h x cdot where mathcal s denotes the solution map recovering the input x 0 from the outputs i x 0 or i x 0 means that these input output maps are injective in the sense that locally x 1 ne x 2 rightarrow i x 1 cap i x 2 emptyset criteria for bo

neural approximations for multistage optimal control of nonlinear stochastic systems two main approximations are used to solve a nonlinear quadratic gaussian lqg optimal control problem the control law is assigned a given structure in which a finite number of parameters have to be determined to minimize the cost function the chosen structure is that of a multilayer feedforward neural network and the control law is given a limited memory the errors resulting front both assumptions are discussed simulation results show that the proposed method may constitute a simple and effective tool for solving to a sufficient degree of accuracy optimal control problems traditionally regarded as difficult ones

state observation by on line minimization a new approach is presented to observing the state of a nonlinear control system the class considered consists of nonlinear observable systems described by an ordinary differential equation in combination with a measurement equation the system functions are required to be twice differentiable based on newton s method an iterative process of state approximation is derived the algorithm computes a piecewise continuous function which is an approximation of the true state the convergence of the proposed algorithm is proved finally several examples show the performance characteristics of the method

moving horizon observers and observer based control in this paper two topics are explored a new approach to the problem of obtaining an estimate of the state of a nonlinear system is proposed the moving horizon observer produces an estimate of the state of the nonlinear system at time t either by minimizing or approximately minimizing a cost function over the preceding interval horizon t t t as t advances so does the horizon convergence of the estimator is established under the assumption that the corresponding global optimization problem can be approximately solved and a uniform reconstructability condition is satisfied the latter condition is automatically satisfied for linear observable systems the utility of the estimator for receding horizon control is explored in particular stability of a composite moving horizon system comprising a moving horizon regulator and a moving horizon observer is established

irregular sampling theorems and series expansions of band limited functions abstract we present a new approach to the problem of irregular sampling of band limited functions that is based on the approximation and factorization of convolution operators a special case of our main result is the following theorem if r n is compact g l 1 rn a band limited function with g t 0 on and xi i i and yj j j are two discrete sets in rn which are dense enough then every band limited function f lwp 1 p our methods work a for a very general class of function spaces not only weighted lwp spaces b in all dimensions even in general locally compact abelian groups c to provide the correct behavior of the coefficients in contrast to the traditional estimates currently used by engineers furthermore the method is constructive and stable with respect to input errors such as round off errors truncation effors jitter errors or aliasing errors

hinging hyperplanes for regression classification and function approximation a hinge function y h x consists of two hyperplanes continuously joined together at a hinge in regression prediction classification pattern recognition and noiseless function approximation use of sums of hinge functions gives a powerful and efficient alternative to neural networks with computation times several orders of magnitude less than is obtained by fitting neural networks with a comparable number of parameters a simple and effective method for finding good hinges is presented

marcinkiewicz zygmund inequalities and interpolation by spherical polynomials with respect to doubling weights abstract we find necessary density conditions for marcinkiewicz zygmund inequalities and interpolation for spaces of spherical polynomials in the weighted l p 0 p space with respect to a doubling weight moreover we show that these conditions are sharp

multi tiling sets riesz bases and sampling near the critical density in lca groups abstract we prove the existence of sampling sets and interpolation sets near the critical density in paley wiener spaces of a locally compact abelian lca group g this solves a problem left by grochenig kutyniok and seip 2008 7 to achieve this result we prove the existence of universal riesz bases of characters for l 2 provided that the relatively compact subset of the dual group g satisfies a multi tiling condition this last result generalizes fuglede s theorem and extends to lca groups setting recent constructions of riesz bases of exponentials in bounded sets of r d

on the connection between exponential bases and certain related sequences in l2 π π a sequence of vectors fn in a separable hilbert space h is a frame if there are positive constants a b such that a f 2 n f fn 2 b f 2 for all f h and fn is a riesz sequence if it is a riesz basis in the closure of the space spanned by the vectors fn the latter of the following two questions has been raised by khrushchev nikolskii and pavlov can every frame of complex exponentials ei nx in l2 be made into a riesz basis by removing from ei nx a suitable collection of the functions ei nx can every riesz sequence ei nx in l2 be made into a riesz basis by adjoining to ei nx a suitable collection of exponentials ei nx ei nx we show that this is indeed so for all frames and riesz sequences ei nx studied so far and we prove that we can always solve both problems in a certain weak sense however our main conclusion is that the answer to both questions is no

some results on the laplacian spread of a graph abstract the laplacian spread of a graph g with n vertices is defined to be s l g 1 g n 1 g where 1 g n 1 g are the largest and the second smallest laplacian eigenvalues of g respectively it is conjectured that s l g n 1 in this paper we first establish a new sharp upper bound for s l g and then use it to prove that the conjecture is true for t quasi regular graphs when t n 3 2 n we also present some other partial solutions for this conjecture in particular we show that the conjecture holds for k 3 free graphs finally we give several sharp lower bounds for s l g as well

comparative study of non linear state observation techniques abstract this paper contains a comparative study of four techniques for observing the slates of non linear systems the first technique examined is inspired by bestle and zeitz and icrener and respondek in this method a non linear transformation is found that brings the system into a canonical form from where observer design is facilitated the second technique is attributable to thau in this method the error between the system s true state and the output of the observer is shown to be asymptotically convergent to zero provided that an additional assumption is valid the third technique is due to baumann and rugh in this method extended or pseudolinearization of the error differential equation about a family of equilibrium points results in an observer design such that the eigenvalues of the linearized error equation are locally invariant finally techniques from variable structure systems are utilized to design an observer that yields an exponentially decaying error like thau s observer but unl

a neural state estimator with bounded errors for nonlinear systems a neural state estimator is described acting on discrete time nonlinear systems with noisy measurement channels a sliding window quadratic estimation cost function is considered and the measurement noise is assumed to be additive no probabilistic assumptions are made on the measurement noise nor on the initial state novel theoretical convergence results are developed for the error bounds of both the optimal and the neural approximate estimators to ensure the convergence properties of the neural estimator a minimax tuning technique is used the approximate estimator can be designed offline in such a way as to enable it to process on line any possible measure pattern almost instantly

beurling density on expansible locally compact groups and systems of translations abstract in this paper we investigate some properties of the expansive automorphisms on expansible locally compact groups we define the upper and lower beurling densities on expansible locally compact groups using this definition we show that if for some 1 p the finite union of translates k 1 n t p f k k is a p bessel sequence then the upper beurling density is finite and if k 1 n t p f k k is a c q system then the upper beurling density cannot be finite in particular we conclude that there exists no p bessel c q system in l p g of the form k 1 n t p f k k where g is an expansible group

neural networks for feedback feedforward nonlinear control systems this paper deals with the problem of designing feedback feedforward control strategies to drive the state of a dynamic system in general nonlinear so as to track any desired trajectory joining the points of given compact sets while minimizing a certain cost function in general nonquadratic due to the generality of the problem conventional methods are difficult to apply thus an approximate solution is sought by constraining control strategies to take on the structure of multilayer feedforward neural networks after discussing the approximation properties of neural control strategies a particular neural architecture is presented which is based on what has been called the linear structure preserving principle the original functional problem is then reduced to a nonlinear programming one and backpropagation is applied to derive the optimal values of the synaptic weights recursive equations to compute the gradient components are presented which generalize the classical adjoint system equations of n stage optimal control theory simulation results related to nonlinear nonquadratic problems show the effectiveness of the proposed method

sup norm approximation bounds for networks through probabilistic methods we consider the problem of approximating a smooth target function and its derivatives by networks involving superpositions and translations of a fixed activation function the approximation is with respect to the sup norm and the rate is shown to be of order o n sup 1 2 that is the rate is independent of the dimension d the results apply to neural and wavelet networks and extend the work of barren see proc 7th yale workshop on adaptive and learning systems may 1992 and ibid vol 39 p 930 1993 the approach involves probabilistic methods based on central limit theorems for empirical processes indexed by classes of functions