spectral methods for weakly singular volterra integral equations with smooth solutions we propose and analyze a spectral jacobi collocation approximation for the linear volterra integral equations vies of the second kind with weakly singular kernels in this work we consider the case when the underlying solutions of the vies are sufficiently smooth in this case we provide a rigorous error analysis for the proposed method which shows that the numerical errors decay exponentially in the infinity norm and weighted sobolev space norms numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence

on the boundedness and continuity of the spectral factorization mapping it is known that the spectral factorization mapping is bounded but not continuous on l infty the space of essentially bounded measurable functions on the unit circle in this article we study the spectral factorization mapping on decomposing banach algebras the most important example of a decomposing banach algebra is the wiener algebra the space of all absolutely convergent fourier series it is shown that the spectral factorization mapping is locally lipschitz continuous but not bounded on all decomposing banach algebras in consideration an application is given to the construction of approximate normalized coprime factorization

spectral method on quadrilaterals in this paper we investigate the spectral method on quadrilaterals we introduce an orthogonal family of functions induced by legendre polynomials and establish some results on the corresponding orthogonal approximation these results play important roles in the spectral method for partial differential equations defined on quadrilaterals as examples of applications we provide spectral schemes for two model problems and prove their spectral accuracy in jacobi weighted sobolev space numerical results coincide well with the analysis we also investigate the spectral method on convex polygons whose solutions possess spectral accuracy the approximation results of this paper are also applicable to other problems

robust identification from band limited data consider the problem of identifying a scalar bounded input bounded output stable transfer function from pointwise measurements at frequencies within a bandwidth we propose an algorithm which consists of building a sequence of maps from data to models converging uniformly to the transfer function on the bandwidth when the number of measurements goes to infinity the noise level to zero and asymptotically meeting some gauge constraint outside error bounds are derived and the procedure is illustrated by numerical experiments

a criterion of integrability for perturbed nonresonant harmonic oscillators wick ordering of the perturbations in classical mechanics and invariance of the frequency spectrum we introduce an analogue to the renormalization theory of quantum fields into classical mechanics we also find an integrability criterion guaranteeing the convergence of birkhoff s series and an algorithm for modifying the hamiltonian to fix the frequency spectrum of the quasi periodic motions we point out its possible relevance to the transition to chaos

robustness of the inner outer factorization and of the spectral factorization for fir data the present paper analyzes the construction of outer functions the factorization of finite impulse response fir systems into a minimum phase system and an all pass part and the spectral factorization it investigates the behavior of these operations with respect to errors in the given data it shows that the error in the constructed outer function grows at least and at most proportional with the logarithm of the degree of the given fir system and proportional to the error in the given data for the other two operations it turns out that they show the same behavior with respect to errors in the given fir data as the construction of outer functions

continuity properties of lqg optimal controllers it is shown that the lqg optimal controller is a continuous function of the plant the result is proved for a class of plants which contains the class of strictly proper finite dimensional plants the topology employed is the one generated by convergence of the closed loop transfer functions in an induced l sense this topology is slightly stronger than the usual h2 gap metric convergence on transfer functions

a counterexample on continuous coprime factors in georgiou and smith 1992 the following question was raised consider a linear shift invariant system on l sub 2 0 spl infin let the graph of the system have fourier transform mn h sup 2 i e the system has a transfer function p n m where m n are elements of c sub a f spl isin h sup spl infin f is continuous on the compactified right half plane is it possible to normalize m and n i e to ensure m sup 2 n sup 2 1 in c sub a the author shows by example that this is not always possible

from a standard factorization to a j spectral factorization for a class of infinite dimensional systems matrix valued functions in the wiener class on the imaginary line are considered in this note this class of functions is large enough to be suitable for many applications in systems and control of infinite dimensional systems for this class of functions three kinds of factorization are discussed right standard factorization canonical factorization and j spectral factorization in particular we focus on an algorithmic procedure to find a right standard factorization and a j spectral factorization for matrix valued functions in the wiener class under the assumption that such factorizations exists in practice the j spectral factors for irrational functions are usually calculated using rational approximations we show that approximation using rational functions may be achieved in the wiener norm

a multistep legendre gauss spectral collocation method for nonlinear volterra integral equations we introduce a multistep legendre gauss spectral collocation method for the nonlinear volterra integral equations of the second kind this method is easy to implement and possesses high order accuracy in addition it is very suitable for long time calculations we also derive the optimal convergence of the hp version of the multistep collocation method under the l 2 norm numerical experiments confirm the theoretical expectations

spectral methods in time for a class of parabolic partial differential equations in this paper we introduce a fully spectral solution for the partial differential equation u t uu x vu x x mu x x x lu x x x x 0 for periodic boundary conditions in space the use of fourier expansion in x admits of a particularly efficient algorithm with respect to expansion of the time dependence in a chebyshev series boundary conditions other than periodic may still be treated with reasonable though lesser efficiency for all cases very high accuracy is attainable at moderate computational cost relative to the expense of variable order finite difference methods in time

an iterative multistep kernel based method for nonlinear volterra integral and integro differential equations of fractional order abstract an iterative multistep kernel based method is proposed for the nonlinear volterra integral equations and nonlinear volterra integro differential equations of fractional order which can produce reliable globally smooth numerical solutions an error estimate of the positive definite kernel interpolation and the convergence and error analysis of the proposed iterative scheme are investigated here we focused on positive definite radial basis kernels and further a new and applicable shape parameter selection strategy is proposed the proposed multi step method set up and solve several small local problems instead of a single large problem which makes it suitable for problems with long time simulations in order to show the efficiency and versatility of the proposed method some numerical experiments are reported the comparison of the numerical results with the analytical solutions and the best reported results in the literature confirm the good accuracy and applicability of the proposed method

quasiperiodic motions in dynamical systems review of a renormalization group approach power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory in the case of quasiperiodic solutions the issue of convergence of the series is bedeviled by the so called small divisor problem in this paper we review a method recently introduced to deal with such a problem based on renormalization group ideas and multiscale techniques applications to both quasi integrable hamiltonian systems kolmogorov arnold moser kam theory and non hamiltonian dissipative systems are discussed the method is also suited to situations in which the perturbation series diverges and a resummation procedure can be envisaged leading to a solution which is not analytic in the perturbation parameter we consider explicitly examples of solutions which are only c in the perturbation parameter or even defined on a cantor set

quasiperiodic hamiltonian motions scale invariance harmonic oscillators the work of kolmogorov arnold and moser appeared just before the renormalization group approach to statistical mechanics was proposed by wilson phys rev b 4 3174 3183 1971 it can be classified as a multiscale approach which also appeared in works on the convergence of fourier s series l carleson acta math 116 135 157 1966 and c fefferman ann math 98 551 571 1973 or construction of euclidean quantum fields e nelson j math phys 5 1190 1964 or the scaling analysis of the short scale behavior of navier stokes fluids l caffarelli r kohn and l nirenberg commun pure appl math 35 771 831 1982 to name a few which resulted in numerous problems in this review the kam theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a trivial fixed point the work of kolmogorov arnold and moser appeared just before the renormalization group approach to statistical mechanics was proposed by wilson phys rev b 4 3174 3183 1971 it can be classified as a multiscale approach which also appeared in works on the convergence of fourier s series l carleson acta math 116 135 157 1966 and c fefferman ann math 98 551 571 1973 or construction of euclidean quantum fields e nelson j math phys 5 1190 1964 or the scaling analysis of the short scale behavior of navier stokes fluids l caffarelli r kohn and l nirenberg commun pure appl math 35 771 831 1982 to name a few which resulted in numerous problems in this review the kam theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a trivial fixed point

spectral factorization for polynomial spectral densities impact of dimension this correspondence investigates the continuity behavior of the spectral factorization mapping for trigonometric polynomials it is clear that this factorization mapping is continuous on the space of all trigonometric polynomials of a fixed degree n which means that a small perturbation in the given spectrum yields always a bounded error in the spectral factor the correspondence derives a lower bound on the continuity constant of the spectral factorization mapping which shows that the error in the spectral factor grows at least proportional with the logarithm of the degree n of the given spectrum

spectral factorization and lq optimal regulation for multivariable distributed systems a necessary and sufficient condition is proved for the existence of a bistable spectral factor with entries in the distributed proper stable transfer function algebra x1d49c in the context of distributed multivariable convolution systems with no delays a by product is the existence of a normalized coprime fraction of the transfer function of such a possibly unstable system with entries in the algebra of fractions over x1d49c we next study semigroup state space systems sgb with bounded sensing and control having a transfer function with entries in and consider its standard lq optimal regulation problem having an optimal state feedback operator k0 for a system sgb a formula is given relating any spectral factor of a transfer function coprime fraction power spectral density to k0 a by product is the description of any normalized coprime fraction of the transfer function in terms of k0 finally we describe an alternative way of finding the solution operator k0 of the lq problem using spectral factor

spectral methods in time for hyperbolic equations a pseudospectral numerical scheme for solving linear periodic hyperbolic problems is described it has infinite accuracy both in time and in space the high accuracy in time is achieved without increasing the computational work and memory space which is needed for a regular one step explicit scheme the algorithm is shown to be optimal in the sense that among all the explicit algorithms of a certain class it requires the least amount of work to achieve a certain given resolution the class of algorithms referred to consists of all explicit schemes which may be represented as a polynomial in the spatial operator

runge kutta theory for volterra integral equations of the second kind the present paper develops the theory of general runge kutta methods for volterra integral equations of the second kind the order conditions are derived by using the theory of p series which for our problem reduces to the theory of v series these results are then applied to two special classes of runge kutta methods introduced by pouzet and by bel tyukov

boundedness behavior of the spectral factorization for polynomial data in the wiener algebra spectral factorization is of fundamental importance in many areas of signal processing this paper investigates the boundedness behavior of the spectral factorization mapping in the wiener algebra thereby the focus lies on the factorization of polynomial spectral densities with a finite degree n since such spectra are especially important for practical applications the paper presents a lower and an upper bound on the boundedness behavior which will show that the boundedness constant of the spectral factorization mapping gets worse as the degree n of the spectra increases therewith one obtains independently the known result that the spectral factorization mapping is unbounded on the wiener algebra

adaptive numerical approach based upon chebyshev operational vector for nonlinear volterra integral equations and its convergence analysis abstract in the current paper we present a direct numerical scheme to approximate a second kind nonlinear volterra integral equations nvies the scheme is based upon shifted chebyshev polynomials and its operational matrices which eventually leads to the sparsity of the coefficients matrix of obtained system the main idea of the proposed approach is based on a useful property of chebyshev polynomials that yields to construct a new operational vector this vector eliminates any requirement of using projection methods and also enhances the accuracy vs other methods applied projection methods the constructive technique and the convergence analysis of this approach under the l w 2 norm are also described numerical experiments and comparisons confirm the applicability and the validity of the presented scheme

degenerate elliptic resonances quasi periodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exist after small perturbations the parametric equations of the invariant tori can often be computed as a formal power series in the perturbation parameter and can be given a meaning via resummations here we prove that for a class of elliptic tori a resummation algorithm can be devised and proved to be convergent thus extending to such lower dimensional invariant tori the methods employed to prove convergence of the lindstedt series either for the maximal i e kam tori or for the hyperbolic lower dimensional invariant tori

hyperbolic low dimensional invariant tori and summations of divergent series we consider a class of a priori stable quasi integrable analytic hamiltonian systems and study the regularity of low dimensional hyperbolic invariant tori as functions of the perturbation parameter we show that under natural nonresonance conditions such tori exist and can be identified through the maxima or minima of a suitable potential they are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin the invariant tori admit an asymptotic series at the origin with taylor coefficients that grow at most as a power of a factorial and a remainder that to any order n is bounded by the n 1 st power of the argument times a power of n we show the existence of a summation criterion of the generically divergent series in powers of the perturbation size that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theory

continuity of the spectral factorization on a vertical strip the continuity of the mapping which associates a spectral factor to a spectral density is investigated this mapping can be defined on several classes of spectral densities and spectral factors for the usual largest class of spectral densities i e essential bounded functions on the imaginary axis that are bounded away from zero it is known that this mapping is not continuous it is shown here that for slightly smaller but still generic class the mapping becomes continuous

kolmogorov arnold moser renormalization group approach to the breakup of invariant tori in hamiltonian systems we analyze the breakup of invariant tori in hamiltonian systems with two degrees of freedom using a combination of kolmogorov arnold moser kam theory and renormalization group techniques we consider a class of hamiltonians quadratic in the action variables that is invariant under the chosen kam transformations following the approach of thirring the numerical implementation of the transformation shows that the kam iteration converges up to the critical coupling at which the torus breaks up by combining this iteration with a renormalization consisting of a shift of resonances and rescalings of momentum and energy we obtain a more efficient method that allows one to determine the critical coupling with high accuracy this transformation is based on the physical mechanism of the breakup of invariant tori we show that the critical surface of the transformation is the stable manifold of codimension one of a nontrivial fixed point and we discuss its universality properties

on the algorithmic solvability of spectral factorization and applications spectral factorization is an operation which appears in many different engineering applications this paper studies whether spectral factorization can be algorithmically computed on an abstract machine a turing machine it is shown that there exist computable spectral densities with very good analytic properties i e smooth with finite energy such that the corresponding spectral factor cannot be determined on a turing machine further it will be proved that it is impossible to decide algorithmically whether or not a given computable density possesses a computable spectral factor this negative result has consequences for applications of spectral factorization in computer aided design because there it is necessary that this problem be decidable conversely this paper will show that if the logarithm of a computable spectral density belongs to certain sobolev space of sufficiently smooth functions then the spectral factor is always computable as an application the paper discusses the possibility of calculating the optimal causal wiener filter on an abstract machine

borel summability and lindstedt series resonant motions of integrable systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a formal power expansion in the strength of the perturbation such series may be sometimes summed via suitable sum rules defining c functions of the perturbation strength here we find sufficient conditions for the borel summability of their sums in the case of two dimensional rotation vectors with diophantine exponent 1 e g with ratio of the two independent frequencies equal to the golden mean

chebyshev spectral solution of nonlinear volterra hammerstein integral equations abstract in this paper the chebyshev spectral cs method for the approximate solution of nonlinear volterra hammerstein integral equations f o k r g r y r d r 0 t is investigated the method is applied to approximate the solution not to the equation in its original form but rather to an equivalent equation z t g t y t t 1 1 the function z is approximated by the nth degree interpolating polynomial zn with coefficients determined by discretizing g t y t at the chebyshev gauss labatto nodes we then define the approximation to y to be of the form y n t f t 1 1 k t s z n s d s 1 1 and establish that under suitable conditions limn yn t y t uniformly in t finally a numerical experiment for a nonlinear volterra hammerstein integral equation is presented which confirms the convergence demonstrates the applicability and the accuracy of the chebyshev spectral cs method

numerical approximations for highly oscillatory bessel transforms and applications abstract this paper presents an efficient numerical method for approximating highly oscillatory bessel transforms based on analytic continuation we transform the integrals into the problems of integrating the forms on 0 with the integrand that does not oscillate and decays exponentially fast which can be efficiently computed by using gauss laguerre quadrature rule we then derive the error of the method depending on the frequency and the node number moreover we apply the scheme for studying the approximations of the solutions of two kinds of highly oscillatory integral equations preliminary numerical results show the efficiency and accuracy of numerical approximations

time discretization of parabolic problems by the hp version of the discontinuous galerkin finite element method the discontinuous galerkin finite element method dgfem for the time discretization of parabolic problems is analyzed in the context of the hp version of the galerkin method error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp dgfem gives spectral convergence in problems with smooth time dependence in conjunction with geometric time partitions it is proved that the hp dgfem results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms for the h version dgfem algebraically graded time partitions are determined that give the optimal algebraic convergence rates a fully discrete hp scheme is discussed exemplarily for the heat equation the use of certain mesh design principles for the spatial discretizations yields exponential rates of convergence in time and space numerical examples confirm the theoretical results

spectral factorization by symmetric extraction for distributed parameter systems the spectral factorization problem of a scalar coercive spectral density is considered in the framework of the callier desoer algebra of distributed parameter system transfer functions criteria are given for the infinite product representation of a meromorphic coercive spectral density of finite order and for the convergence of infinite product representations of spectral factors i e for the convergence of the symmetric extraction method for solving the spectral factorization problem of such spectral density these convergence criteria are applied to the solution of the linear quadratic optimal control problem by spectral factorization for a specific class of semigroup hilbert state space systems with a riesz spectral generator the speed of convergence of the symmetric extraction method is also considered as an example a damped vibrating string model is handled

parallel in time algorithm with spectral subdomain enhancement for volterra integral equations nsf of china 11201077 11071203 91130002 nsf of fujian province 2012j01007 fuzhou university 0460022456 hong kong research grant council gif grants hong kong baptist university frg grants

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

an h p version of the continuous petrov galerkin finite element method for volterra integro differential equations with smooth and nonsmooth kernels we present an h p version of the continuous petrov galerkin cpg finite element method for linear volterra integro differential equations with smooth and nonsmooth kernels we establish a priori error estimates in the l 2 h 1 and l infty norms that are completely explicit with respect to the local discretization and regularity parameters for singular solutions caused by the weakly singular kernels we prove that algebraic convergence rates can be achieved for the h p version of the cpg method with quasi uniform meshes moreover we show that exponential rates of convergence can be achieved for solutions with start up singularities by using geometric time partitions and linearly increasing polynomial degrees numerical experiments are provided to illustrate the theoretical results

kam theorem and quantum field theory we give a new proof of the kam theorem for analytic hamiltonians the proof is inspired by a quantum field theory formulation of the problem and is based on a renormalization group argument treating the small denominators inductively scale by scale the crucial cancellations of resonances are shown to follow from the ward identities expressing the translation invariance of the corresponding field theory

approximative model reconstruction of cascade systems this letter considers how to approximately reconstruct a cascade system from a given unstructured system estimate many system identification methods including subspace methods provide reliable but generally unstructured black box models the problem we consider is how to find cascade systems that are close to such black box models for this we use model matching techniques and optimal weighted hankel norm approximation to obtain accurate low order cascade systems we show that it is possible to bound the reconstruction error in terms of an error tolerance parameter and weighted hankel singular values the suggested methods are illustrated on both a numerical example and a real double tank system with experimental data

convergence analysis of the jacobi spectral collocation methods for volterra integral equations with a weakly singular kernel in this paper a jacobi collocation spectral method is developed for volterra integral equations of second kind with a weakly singular kernel we use some function transformation and variable transformations to change the equation into a new volterra integral equation deflned on the standard interval i1 1 so that the solution of the new equation possesses better regularity and the jacobi orthogonal polynomial theory can be applied conveniently in order to obtain high order accuracy for the approx imation the integral term in the resulting equation is approximated by using jacobi spectral quadrature rules the convergence analysis of this novel method is based on the lebesgue constants corresponding to the lagrange interpolation polynomials poly nomials approximation theory for orthogonal polynomials and the operator theory the spectral rate of convergence for the proposed method is established in the l 1 norm and weighted l 2 norm numerical results are presented to demonstrate the efiectiveness of the proposed method

preserving stability implicit euler method for nonlinear volterra and neutral functional differential equations in banach space this paper is concerned with the contractivity and asymptotic stability properties of the implicit euler method iem for nonlinear functional differential equations fdes these properties are first analyzed for volterra fdes and then the analysis is extended to the case of neutral fdes nfdes such an extension is particularly important since nfdes are more general and have received little attention in the literature the main result we establish is that the iem with linear interpolation can completely preserve these stability properties of the analytical solution to such fdes

on the recovery of functions from pointwise boundary values in a hardy sobolev class of the disk abstract this paper deals with a particular problem in convergent interpolation to analytic functions from boundary values we first stress some system theoretic motivations for these questions then a linear triangular interpolation scheme for analytic functions in the hardy sobolev class d 2 of the disk is studied when the interpolation points lie on the boundary circle specifically the sequence of interpolating functions of minimum norm is shown to converge to the original function provided the closure of the interpolation set has positive measure on the circle this induces uniform convergence in the sense of the usual hardy norms which is further estimated from above and from below in the particular case where the interpolation points are dense in some subarc of the circle