semigroup structural form for bernstein type operators and its applications in this note we provide a semigroup viewpoint for bernstein type operators a main ingredient of our approach is to obtain some explicit semigroup representation formulae which as simple forms are proposed for the fist time as direct applications we improve the corresponding results of de la cal and luquin 1992 1994 moreover some kinds of voronovskaja type results are made by using general semigroup expansion in the end we discuss corresponding generalizations of integral operators with some examples of hybrid durrmeyer type operators

energy decay for a damped nonlinear coupled system this paper proves the well posedness and uniform stabilization of a nonlinear coupled system we estimate the energy decay rate by using the multiplier method

positive definite toeplitz matrices the arnoldi process for isometric operators and gaussian quadrature on the unit circle abstract we show that the well known levinson algorithm for computing the inverse cholesky factorization of positive definite toeplitz matrices can be viewed as a special case of a more general process the latter process provides a very efficient implementation of the arnoldi process when the underlying operator is isometric this is analogous with the case of hermitian operators where the hessenberg matrix becomes tridiagonal and results in the hermitian lanczos process we investigate the structure of the hessenberg matrices in the isometric case and show that simple modifications of them move all their eigenvalues to the unit circle these eigenvalues are then interpreted as abscissas for analogs of gaussian quadrature now on the unit circle instead of the real line the trapezoidal rule appears as the analog of the gauss legendre formula

rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle we prove several results about zeros of paraorthogonal polynomials using the theory of rank one perturbations of unitary operators in particular we obtain new details on the interlacing of zeros for successive popuc

global existence and asymptotic behavior for a coupled hyperbolic system with localized damping abstract in this paper we investigate the global existence and energy decay rate for the solution of a coupled hyperbolic system the semi explicit energy decay rate is established by using piecewise multiplier techniques and weighted integral inequality we extend the energy decay result in alabau boussouira f alabau boussouira convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems appl math optim 51 2005 61 105 for a single equation to the coupled hyperbolic system

on convergence rates in approximation theory for operator semigroups abstract we create a new functional calculus approach to approximation formulas for c 0 semigroups on banach spaces restricted to the domains of fractional powers of their generators this approach allows us to equip the approximation formulas with rates which appear to be optimal in a natural sense in the case of analytic semigroups we improve our general results obtaining better convergence rates which are optimal in that case too the setting of analytic semigroups includes also the case of convergence on the whole space as an illustration of our approach we deduce optimal convergence rates in classical approximation formulas for c 0 semigroups restricted to the domains of fractional powers of their generators

decay estimates for some semilinear damped hyperbolic problems let be a bounded open domain in rn g r r a non decreasing continuous function such that g 0 0 and h e lloc1 r l2 under suitable assumptions on g and h the rate of decay of the difference of two solutions is studied for some abstract evolution equations of the general form u lu g u h t x as t the results obtained by use of differential inequalities can be applied to the case of the semilinear wave equation u u delta u g text u t text h text in r times omega text u text 0 on r times partial omega in r u 0 onr for instance if g s c left s right p 1 s d left s right q 1 s with c d 0 and 1 p q n 2 q n 2 then if h in l infty r l 2 omega all solutions are bounded in the energy space for t 0 and if u v are two such solutions the energy norm of u t v t decays like t 1 p 1 as t

convergence of derivatives for certain mixed szasz beta operators in this paper we study the mixed summation integral type operators having szasz and beta basis functions in summation and integration respectively we obtain the rate of point wise convergence a voronovskaja type asymptotic formula and an error estimate in simultaneous approximation

real orthogonal polynomials in frequency analysis univ estadual paulista ibilce dept ciencias computacao estatist br 15054000 sao paulo brazil

anti szego quadrature rules szego quadrature rules are discretization methods for approximating integrals of the form f e t du t this paper presents a new class of discretization methods which we refer to as anti szego quadrature rules anti szego rules can be used to estimate the error in szego quadrature rules under suitable conditions pairs of associated szego and anti szego quadrature rules provide upper and lower bounds for the value of the given integral the construction of anti szego quadrature rules is almost identical to that of szego quadrature rules in that pairs of associated szego and anti szego rules differ only in the choice of a parameter of unit modulus several examples of szego and anti szego quadrature rule pairs are presented

uniform stabilization of the wave equation by nonlinear boundary feedback the question of uniformly stabilizing the solution of the wave equation y delta y 0 in omega times 0 infty omega is a bounded domain of bf r n by means of a nonlinear feedback law of the following form is studied partial y partial v k x g y on gamma 0 times 0 infty y 0 on gamma 1 times 0 infty gamma 0 gamma 1 being a suitable partition of the boundary of omega and g a continuous nondecreasing function such that g 0 0 we choose k x in l infty k x geqq 0 such that k x vanishes linearly at the interface points x in bar gamma 0 cap bar gamma 1 then if g s behaves like s p 1 s as s to 0 with p 1 and linearly as s to infty it is proved that the energy of every solution decays like t 2 p 1 as t to infty in the case where p 1 the exponential decay rate is proved

existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term the wave equation with a source term is considered utt u u uin 0 we prove the existence and uniform decay rates of the energy by assuming a nonlinear feedback ut acting on the boundary provided that has necessarily not a polynomial growth near the origin to obtain the existence of global solutions we make use of the potential well method combined with the faedo galerkin procedure and constructing a special basis furthermore we prove that the energy of the system decays uniformly to zero and we obtain an explicit decay rate estimate adapting the ideas of lasiecka and tataru differential integral equations 6 3 1993 507 and patrick martinez esain control optimisation calc var 4 1999 419 the resulting problem generalizes martinez results and complements the works of lasiecka and tataru 1993 and vitillaro glasgow math j 44 2002 375

approximating szasz and gamma operators by baskakov operators abstract it is known that both szasz and gamma operators are limits in an appropriate sense of baskakov operators in this paper we obtain rates of convergence for these approximations we also show that convergences are monotonic if the involved function is convex

direct estimates for certain szasz type operators we deal with the modified forms of the szasz basis functions we introduce a new sequence of linear positive operators of durrmeyer type direct estimates for the new sequence have been discussed an open problem is also proposed the present paper deals with the modified forms of the szasz basis functions we propose a durrmeyer type operator having the basis functions in summation and integration due to jain 1972 and pa lta nea 2008 we establish some direct results which include the asymptotic formula and error estimation in terms of the modulus of continuity and weighted approximation in the end we give an open problem for the readers

orthogonal polynomials and quadrature rules on the unit circle associated with perturbations of symmetric measures abstract it was shown recently that given a pair of real sequences c n n 1 d n n 1 with d n n 1 a positive chain sequence we can associate a unique nontrivial probability measure on the unit circle and conversely in this paper we consider the set q c of all nontrivial probability measures for which c n 1 n c with c r we show that every measure q c can be obtained from a perturbation of a symmetric measure on 1 1 moreover the sequence of orthogonal polynomials associated with can be given in terms of a perturbation of symmetric orthogonal polynomials associated with we also prove that every measure in q c is quasi symmetric that is there exists a complex valued function q c z satisfying d z q c z d z and such that q c z 1 when c 0 quadrature rules with quasi symmetric weights are also considered finally some examples of orthogonal polynomials on the unit circle and its associated quasi symmetric nontrivial probability measures are obtained

rate of convergence for certain families of summation integral type operators in this paper we study the rate of convergence for the iterative combinations of a certain family of linear positive operators in terms of higher order integral modulus of smoothness we have used the technique of linear approximating method namely steklov mean to prove the main result in the end we propose some other sequences of linear positive operators and obtain the recurrence formulae for the central moments voronovskaja type asymptotic formulae and error estimations

energy decay rate for a coupled hyperbolic system with nonlinear damping abstract in this paper we investigate the energy decay rate for the solution of a coupled hyperbolic system the explicit energy decay rate is established by using multiplier techniques and constructing a suitable energy functional

well posedness and optimal decay rates for the wave equation with nonlinear boundary damping source interaction we establish subject to some natural additional assumptions imposed on the relation between the source and the damping both well posedness and effective optimal decay rates for the solutions of a semilinear model of the wave equation the theory presented allows to consider both superlinear and sublinear behaviours of the dissipation in the presence of unstructured sources

positive quadrature formulas iii asymptotics of weights first we discuss briefly our former characterization theorem for positive interpolation quadrature formulas abbreviated qf provide an equivalent characterization in terms of jacobi matrices and give links and applications to other qf in particular to gauss kronrod quadratures and recent rediscoveries then for any polynomial t n which generates a positive qf a weight function depending on n is given with respect to which t n is orthogonal to p n 1 with the help of this result an asymptotic representation of the quadrature weights is derived in general the asymptotic behaviour is different from that of the gaussian weights only under additional conditions do the quadrature weights satisfy the so called circle law corresponding results are obtained for positive qf of radau and lobatto type

general decay rate estimates for viscoelastic dissipative systems abstract the linear viscoelastic equation is considered we prove uniform decay rates of the energy by assuming a nonlinear feedback acting on the boundary without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function our estimate depends both on the behavior of the damping term near zero and on the behavior of the relaxation function at infinity the proofs are based on the multiplier method and on a general lemma about convergent and divergent series for obtaining the uniform decay rates

equality of the spectral and dynamical definitions of reflection for full line jacobi matrices schrodinger operators and cmv matrices we show that being reflectionless in the sense of the well known property of m functions is equivalent to a lack of reflection in the dynamics in the sense that any state that goes entirely to x as t goes entirely to x as t this allows us to settle a conjecture of deift and simon from 1983 regarding ergodic jacobi matrices

energy decay rates for the timoshenko system of thermoelastic plates abstract we consider a model describing nonlinear dynamical motions of an unbounded thermoelastic plate we prove the well posedness of the above system and analyse the behaviour of the total energy e t as t our main result shows that the total energy of the system satisfies the following estimate there exist a constant e 0 0 such that e t 4 e 0 exp t for all t 0 the result is proved by constructing a lyapunov function which is a suitable perturbation of the energy and satisfies a differential inequality leading to the desired decay estimate

generalized baskakov szasz type operators abstract in the present paper we introduce generalized baskakov szasz type operators and study some approximation properties of these operators e g rate of convergence in ordinary and simultaneous approximation statistical convergence and the estimate of the rate of convergence for absolutely continuous functions having a derivative coinciding a e with a function of bounded variation

a connection between quadrature formulas on the unit circle and the interval 1 1 abstract we establish a relation between gauss quadrature formulas on the interval 1 1 that approximate integrals of the form i f 1 1 f x x d x and szeg quadrature formulas on the unit circle of the complex plane that approximate integrals of the form i f f e i d the weight x is positive on 1 1 while the weight is positive on it is shown that if cos sin then there is an intimate relation between the gauss and szeg quadrature formulas moreover as a side result we also obtain an easy derivation for relations between orthogonal polynomials with respect to x and orthogonal szeg polynomials with respect to inclusion of gauss lobatto and gauss radau formulas is natural

szego quadrature formulas for certain jacobi type weight functions in this paper we are concerned with the estimation of integrals on the unit circle of the form 02 f ei d by means of the so called szego quadrature formulas i e formulas of the type j 1n jf xj with distinct nodes on the unit circle exactly integrating laurent polynomials in subspaces of dimension as high as possible when considering certain weight functions related to the jacobi functions for the interval 1 1 nodes xj j 1n and weights j j 1n in szego quadrature formulas are explicitly deduced illustrative numerical examples are also given

sieved para orthogonal polynomials on the unit circle abstract sieved orthogonal polynomials on the unit circle were introduced independently by ismail and li 1992 15 and marcellan and sansigre 1991 19 we look at the para orthogonal polynomials chain sequences and quadrature formulas that follow from the kernel polynomials of sieved orthogonal polynomials on the unit circle

szegő lobatto quadrature rules gauss type quadrature rules with one or two prescribed nodes are well known and are commonly referred to as gauss radau and gauss lobatto quadrature rules respectively efficient algorithms are available for their computation szego quadrature rules are analogs of gauss quadrature rules for the integration of periodic functions they integrate exactly trigonometric polynomials of as high degree as possible szego quadrature rules have a free parameter which can be used to prescribe one node this paper discusses an analog of gauss lobatto rules i e szego quadrature rules with two prescribed nodes we refer to these rules as szego lobatto rules their properties as well as numerical methods for their computation are discussed

long time behavior of a coupled heat wave system arising in fluid structure interaction this paper is devoted to the analysis of the long time behavior of a coupled wave heat system in which a wave and a heat equation evolve in two bounded domains with natural transmission conditions at a common interface these conditions couple in particular the heat unknown with the velocity of the wave solution this model may be viewed as a simplified version of linearized models which arise in fluid structure interaction first we show the strong asymptotic stability of solutions to this system then based on the construction of ray like solutions by means of geometric optics expansions and a careful analysis of the transfer of energy at the interface we show the lack of uniform decay in general domains further we obtain a polynomial decay result for smooth solutions of the system under a suitable geometric assumption which guarantees that the heat domain envelopes the wave domain finally in the absence of geometric conditions we show a logarithmic decay result for the same system but with simplified transmission conditions at the interface we also analyze the difficulty there is to extend this result to the more natural transmission conditions

quadratic optimization for infinite dimensional linear differential difference type systems syntheses via the fredholm equation a closed loop solution for quadratic optimization is presented for control systems whose model is given by infinite dimensional linear differential difference equations frac dx t dt ax t sum i 1 n a i x left t h i right bu t sum j 1 m b j u left t r j right a nonsemigroup evolution formula for the fundamental solution is given first the system is then reduced to a volterra integral system by the semicausality approach a new optimality principle is established and the closed loop optimal control is given by real time state feedback plus retarded state integral feedback the feedback operator is determined by solving a linear fredholm integral equation the two crucial methods semicausal dynamical optimization for volterra integral systems and nonsemigroup evolution property are brought together to generalize the syntheses results to infinite dimensional cases

cmv matrices with asymptotically constant coefficients szego blaschke class scattering theory abstract we develop a modern extended scattering theory for cmv matrices with asymptotically constant verblunsky coefficients we show that the traditional faddeev marchenko condition is too restrictive to define the class of cmv matrices for which there exists a unique scattering representation the main results are 1 the class of twosided cmv matrices acting in l 2 whose spectral density satisfies the szego condition and whose point spectrum the blaschke condition corresponds precisely to the class where the scattering problem can be posed and solved that is to a given cmv matrix of this class one can associate the scattering data and the fm space the cmv matrix corresponds to the multiplication operator in this space and the orthonormal basis in it corresponding to the standard basis in l 2 behaves asymptotically as the basis associated with the free system 2 from the point of view of the scattering problem the most natural class of cmv matrices is that one in which a the scattering data determine the matrix uniquely and b the associated gelfand levitan marchenko transformation operators are bounded necessary and sufficient conditions for this class can be given in terms of an a 2 kind condition for the density of the absolutely continuous spectrum and a carleson kind condition for the discrete spectrum similar conditions close to the optimal ones are given directly in terms of the scattering data

long time behavior of solutions to a nonlinear hyperbolic relaxation system abstract we study the large time behavior of solutions of a one dimensional hyperbolic relaxation system that may be written as a nonlinear damped wave equation first we prove the global existence of a unique solution and their decay properties for sufficiently small initial data we also show that for some large initial data solutions blow up in finite time for quadratic nonlinearities we prove that the large time behavior of solutions is given by the fundamental solution of the viscous burgers equation in some other cases the convection term is too weak and the large time behavior is given by the linear heat kernel

decay estimates for second order evolution equations with memory abstract this paper develops a unified method to derive decay estimates for general second order integro differential evolution equations with semilinear source terms depending on the properties of convolution kernels at infinity we show that the energy of a mild solution decays exponentially or polynomially as t our approach is based on integral inequalities and multiplier techniques these decay results can be applied to various partial differential equations we discuss three examples a semilinear viscoelastic wave equation a linear anisotropic elasticity model and a petrovsky type system

quadrature and orthogonal rational functions abstract classical interpolatory or gaussian quadrature formulas are exact on sets of polynomials the szeg quadrature formulas are the analogs for quadrature on the complex unit circle here the formulas are exact on sets of laurent polynomials in this paper we consider generalizations of these ideas where the laurent polynomials are replaced by rational functions that have prescribed poles these quadrature formulas are closely related to certain multipoint rational approximants of cauchy or riesz herglotz transforms of a positive or general complex measure we consider the construction and properties of these approximants and the corresponding quadrature formulas as well as the convergence and rate of convergence

analytic continuation of eigenvalues of the lame operator abstract eigenvalues of the lame operator are studied as complex analytic functions in period of an elliptic function we investigate the branching of eigenvalues numerically and clarify the relationship between the branching of eigenvalues and the convergent radius of a perturbation series

trigonometric orthogonal systems and quadrature formulae quadrature rules with maximal even trigonometric degree of exactness are considered we give a brief historical survey on such quadrature rules special attention is given on an approach given by turetzkii a h turetzkii on quadrature formulae that are exact for trigonometric polynomials east j approx 11 3 2005 337 359 translation in english from uchenye zapiski vypusk 1 149 seria math theory of functions collection of papers izdatel stvo belgosuniversiteta imeni v i lenina minsk 1959 pp 31 54 the main part of the topic is orthogonal trigonometric systems on 0 email protected or on email protected p with respect to some weight functions w x we prove that the so called orthogonal trigonometric polynomials of semi integer degree satisfy a five term recurrence relation in particular we study some cases with symmetric weight functions also we present a numerical method for constructing the corresponding quadratures of gaussian type finally we give some numerical examples also we compare our method with other available methods

linear combinations of orthogonal polynomials generating positive quadrature formulas let pk x x k e n0 be the polynomials orthogonal on 1 1 with respect to the positive measure da we give sufficient conditions on the real numbers p j 0 m such that the linear combination of orthogonal polynomials yflo jpn j has n simple zeros in 1 1 1 and that the interpolatory quadrature formula whose nodes are the zeros of yfj osjpn j has positive weights

locally distributed control and damping for the conservative systems in this paper we note the equivalence between exact controllability and exponential stabilizability for an abstract conservative system with bounded control this enables us to establish a frequency domain characterization for the exact controllability uniform exponential decay property of second order elastic systems such as the wave equation and the petrovsky equation with locally distributed control damping a piecewise multiplier method for frequency domain is introduced for several classes of pdes on regions which are not necessarily smooth we obtain a sufficient condition for the subregion on which the application of control damping will yield the exact controllability uniform exponential decay property this result provides useful information for designing the location of controllers dampers for distributed systems with a law of conservation

perturbations on the subdiagonals of toeplitz matrices let l be an hermitian linear functional defined on the linear space of laurent polynomials it is very well known that the gram matrix of the associated bilinear functional in the linear space of polynomials is a toeplitz matrix in this contribution we analyze some linear spectral transforms of l such that the corresponding toeplitz matrix is the result of the addition of a constant in a subdiagonal of the initial toeplitz matrix we focus our attention in the analysis of the quasi definite character of the perturbed linear functional as well as in the explicit expressions of the new monic orthogonal polynomial sequence in terms of the first one