the extended krylov subspace method and orthogonal laurent polynomials abstract the need to evaluate expressions of the form f a v where a is a large sparse or structured symmetric matrix v is a vector and f is a nonlinear function arises in many applications the extended krylov subspace method can be an attractive scheme for computing approximations of such expressions this method projects the approximation problem onto an extended krylov subspace k l m a span a l 1 v a 1 v v a v a m 1 v of fairly small dimension and then solves the small approximation problem so obtained we review available results for the extended krylov subspace method and relate them to properties of laurent polynomials the structure of the projected problem receives particular attention we are concerned with the situations when m l and m 2 l

extremal solutions of the strong stieltjes moment problem abstract a solution of the strong stieltjes moment problem for the sequence cn n o 1 2 is a finite positive measure on 0 such that c n 0 t n d t for all n while a solution of the strong hamburger moment problem for the same sequence is a finite positive measure on such that c n t n d t for all n when the hamburger problem is indeterminate there exists a one to one correspondence between all solutions and all nevanlinna functions the constant included the correspondence is given by f z z z z z z z where are certain functions holomorphic in c 0 the extremal solutions are the solutions t corresponding to the constant functions z t t r the accumulation points of the isolated set zt of zeros of z t z consists of 0 and the support of the extremal solution t is the set zt 0 there exists an interval t 0 t such that the extremal solutions of the stieltjes problem are exactly those t for which t t 0 t the measures t 0 and t are natural solutions and the only ones if k n denote the zeros of the orthogonal laurent polynomials determined by cn ordered by size then k n tends to 0 and n k n tends to for arbitrary constant k when n tends to

moment problems and orthogonal functions abstract the caratheodory coefficient problem for an infinite sequence c n can be formulated as follows find a caratheodory function f z i e an analytic function mapping the unit disk d z z into the right half plane p w re w 0 whose taylor coefficients at a fixed points in d for convenience the origin are the c n this problem is equivalent to the trigonometric moment problem find a measure on such that e i n d c n for n 0 1 2 these problems are closely related to the theory of szego polynomials i e orthogonal polynomials on the unit circle t z z 1 let n be an arbitrary given sequence of not necessarily distinct points in d and let w n be a given sequence of numbers the nevanlinna pick problem for this situation is to find a caratheodory function f z with the interpolation property f n w n for all n when points n are repeated the interpolation property involves the appropriate number of taylor coefficients also this problem is connected with the problem of finding a measure generating moments the nevanlinna pick problem is related to a theory of rational functions which are orthogonal on t with poles at the points 1 n this relationship is analogous to and generalizes the relationship between the caratheodory coefficient problem or trigonometric moment problem and polynomials orthogonal on t the situation when points n are given on the boundary t or d is also briefly discussed

a continuum limit of the relativistic toda lattice asymptotic theory of discrete laurent orthogonal polynomials with varying recurrence coefficients we consider the continuum limit of the relativistic toda lattice in particular we propose a method in order to integrate this system of nonlinear partial differential equations for some particular initial data and boundary conditions before possible shocks first we recall the relation between the finite relativistic toda lattice and the theory of discrete laurent orthogonal polynomials our analysis is then based on some results for the asymptotic theory of discrete laurent orthogonal polynomials with varying recurrence coefficients and the connection with a constrained and weighted extremal problem for logarithmic potentials

convergence of two point pade approximants to series of stieltjes abstract in this paper we study the convergence of two point pade approximants to stieltjes series a carleman type condition given by gragg 1980 in the context of convergence theory for continued fractions is here established in the framework of orthogonal laurent polynomials jones njastad and thron 1984 jones and thron 1981 njastad and thron 1983 1986 a bound for the error is also given

orthogonal laurent polynomials and strong moment theory a survey these topics are found in many parts of 20th century mathematics and its applications in mathematical physics chemistry statistics and engineering historically the analytic theory of continued fractions has played a central role in both the origin and the development of the other topics continued fractions are intimately related to pad e approximants and special functions emphasis is given to the development of strong moment theory and orthogonal laurent polynomials and to the related continued fractions quadrature formulas integral transforms and linear functionals by a strong moment problem we mean the following for a given bisequence n n of real numbers does there exist a distribution function such that

quadrature formulas on the unit circle based on rational functions abstract quadrature formulas on the unit circle were introduced by jones in 1989 on the other hand bultheel also considered such quadratures by giving results concerning error and convergence in other recent papers a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well known szego polynomials in this paper these quadratures are again analyzed and results about convergence given furthermore an application to the poisson integral is also made

gaussian quadrature rules and numerical examples for strong extensions of mass distribution functions abstract the theory of strong moment problems has provided gaussian quadrature rules for approximate integration with respect to strong distributions in hagler ph d thesis university of colorado boulder 1997 and hagler et al lecture notes in pure and applied mathematics marcel dekker new york in press a transformation of the form v x 1 x x 0 is used to obtain strong mass distribution functions from mass distribution functions this transformation also links the systems of orthogonal polynomials and laurent polynomials and their zeros in this paper we show how the transformation method can be used to obtain the gaussian quadrature rules for strong extensions of mass distribution functions we then provide numerical examples of strong gaussian quadrature approximations to the integrals of elementary functions with respect to selected strong distributions

brief paper adaptive output control of uncertain nonlinear systems with non symmetric dead zone input this paper deals with the adaptive output feedback control problem of a class of uncertain nonlinear systems with an unknown non symmetric dead zone nonlinearity the nonlinear system considered here is dominated by a triangular system without zero dynamics satisfying polynomial growth in the unmeasurable states an adaptive control scheme is developed without constructing the dead zone inverse the proposed adaptive control scheme requires only the information of bounds of the slopes and the breakpoint of dead zone nonlinearity the novelty of this paper is that a universal type adaptive output feedback controller is numerically constructed by using a sum of squares sos optimization algorithm which ensures the boundedness of all the signals in the adaptive closed loop without knowing the growth rate of the uncertainties an example is presented to show the effectiveness of the proposed approach

extremal properties of strong quadrature weights and maximal mass results for truncated strong moment problems abstract a bisequence of complex numbers n determines a strong moment functional l satisfying l xn n if l is positive definite on a bounded interval a b r 0 then l has an integral representation l n 0 1 2 and quadrature rules wni xni exist such that k i innsnikwni this paper is concerned with establishing certain extremal properties of the weights wni and using these properties to obtain maximal mass results satisfied by distributions x representing l when only a finite bisequence of moments k k nn 1 is given

adaptive backstepping controller design using stochastic small gain theorem a more general class of stochastic nonlinear systems with unmodeled dynamics and uncertain nonlinear functions are considered in this paper with the concept of input to state practical stability isps and nonlinear small gain theorem being extended to stochastic case by combining stochastic small gain theorem with backstepping design technique an adaptive output feedback controller is proposed it is shown that the closed loop system is practically stable in probability a simulation example demonstrates the control scheme

orthogonal laurent polynomials and continued fractions associated with log normal distributions abstract this paper describes properties and computational procedures related to orthogonal laurent polynomials continued fractions two point pade approximants strong moment problems and l gaussian quadrature associated with log normal distribution functions t defined by t q1 2 2 e lnt 2 2 0

adaptive dead zone compensation for output feedback canonical systems an adaptive output feedback dead zone compensation scheme is designed for systems with an unknown dead zone at the input of an n th order smooth nonlinear dynamics in the output feedback canonical form the proposed adaptive controller employs an adaptive dead zone inverse to cancel the dead zone and uses a backstepping design for adaptive output feedback control it has a new state observer parametrization that is needed to handle the dead zone uncertainty and a new robust control law that is suitable for the parameter projection needed to implement an adaptive dead zone inverse the adaptive dead zone compensation design ensures closed loop signal boundedness and improves system tracking performance

adaptive fuzzy robust output feedback control of nonlinear systems with unknown dead zones based on a small gain approach in this paper an adaptive fuzzy robust output feedback control problem is considered for a class of single input and single output nonlinear systems in a strict feedback form the considered systems possess the unstructured uncertainties unknown dead zone and the dynamics uncertainties and they do not assume the states being available for the controller design in the controller design fuzzy logic systems are first used to approximate the unstructured uncertainties and by utilizing the information of the bounds of the dead zone slopes and treating the time varying inputs coefficients as a system uncertainty a fuzzy state observer is designed to estimate the unmeasured states by combining a backstepping technique with a nonlinear small gain approach a new adaptive fuzzy robust output feedback control has been developed it is proved that the proposed fuzzy adaptive control approach can guarantee the semiglobal uniform ultimate boundedness for all the solutions of the closed loop systems simulation studies and comparisons with previous methods are included to illustrate the effectiveness of the proposed approach

a modified schur algorithm and an extended hamburger moment problem an algorithm for a pick nevanlinna problem where the interpola tion points coalesce into a finite set of points on the real line is introduced its connection with certain multipoint pade approximation problems is discussed and the results are used to obtain the solutions of an extended hamburger mo ment problem the pick nevanlinna problem is the following let za a a be a set of distinct points in the open upper half plane h z imz 0 and let wa a a be a set of points in c find a function f z which is analytic in such that imf z 0 for z e j and f z wa for all a a a function f z which is analytic for z hq with imf z 0 is called a nevanlinna function the problem was solved by pick 27 28 in the case that a is finite by nevanlinna 15 16 in the case that a is countable and by krein and rekhtman 14 in the general case a variant of the problem for finite or countable a arises when all points za coalesce to a single point a and given values wa at the points are replaced by the taylor coefficients at a a problem closely related to this is caratheodory s coefficient problem see 2 3 33 given a finite sequence y0 ym or an infinite sequence yn n n find a function f z which is analytic in the open unit disc d z z 0 for z d and f z ezu 2 ezm okm zk or f z er o z for historical remarks on this problem see 12 by introducing the linear fractional transformation 7 j we can refor mulate the requirement re7 z 0 to read 7 z 0 for z d or a schur function if 7 z 1 for z e 7j schur 31 invented an algorithm called the schur algorithm to deal with this problem the tech nique was adapted by nevanlinna to deal with the pick nevanlinna problem

adaptive fuzzy output feedback control of mimo nonlinear systems with unknown dead zone inputs this paper is concerned with the problem of adaptive fuzzy tracking control for a class of multi input and multi output mimo strict feedback nonlinear systems with both unknown nonsymmetric dead zone inputs and immeasurable states in this research fuzzy logic systems are utilized to evaluate the unknown nonlinear functions and a fuzzy adaptive state observer is established to estimate the unmeasured states based on the information of the bounds of the dead zone slopes as well as treating the time varying inputs coefficients as a system uncertainty a new adaptive fuzzy output feedback control approach is developed via the backstepping recursive design technique it is shown that the proposed control approach can assure that all the signals of the resulting closed loop system are semiglobally uniformly ultimately bounded it is also shown that the observer and tracking errors converge to a small neighborhood of the origin by selecting appropriate design parameters simulation examples are also provided to illustrate the effectiveness of the proposed approach

orthogonal laurent polynomials and two point pade approximants associated with dawson s integral starting from the coefficients of the power expansions for the dawson s integral both around the origin and infinity a linear functional acting on the space of laurent polynomials is defined in this paper properties of this functional are studied in connection with certain sequences of orthogonal laurent polynomials and two point pade approximants

on the ruin probability for nonhomogeneous claims and arbitrary inter claim revenues recently raducan et al 2015 obtained recursive formulas for the ruin probability of a surplus process at or before claim instants under the assumptions that the claim sizes are independent nonhomogeneous erlang distributed and independent of the inter claim times i e the times between two successive claims which are assumed to be independent identically distributed i i d following an erlang or a mixture of exponentials distribution in this paper we extend these formulas to the more general case when the inter claim times are i i d nonnegative random variables following an arbitrary distribution we also present numerical results based on the new recursions discuss some computational aspects and state a conjecture that connects the ordering of the claims arrival with the magnitude of the corresponding ruin probabilities

adaptive fuzzy output feedback tracking backstepping control of strict feedback nonlinear systems with unknown dead zones in this paper an adaptive fuzzy backstepping control approach is considered for a class of nonlinear strict feedback systems with unknown functions unknown dead zones and immeasurable states fuzzy logic systems are utilized to approximate the unknown nonlinear functions and a fuzzy filters state observer is designed to estimate the immeasurable states by using the adaptive backstepping recursive design technique and constructing the dead zone inverse a new adaptive fuzzy backstepping output feedback control approach is developed it is mathematically proved that all the signals of the resulting closed loop adaptive control system are semiglobally uniformly ultimately bounded and the tracking error converges to a small neighborhood of the origin by appropriate choice of design parameters the proposed approach cannot only solve the problem of the dead zones but also cancel the restrictive assumption in the previous literature that the states are all available for measurement two simulation examples are provided to show the effectiveness of the proposed approach

joint density of the number of claims until ruin and the time to ruin in the delayed renewal risk model with erlang n claims in this paper we consider a delayed renewal risk model where the first inter arrival time can be an arbitrary positive random variable while the following inter arrival times are assumed to be infinitely divisible and further the claim sizes are erlang distributed applying the approach of borovkov and dickson 2008 7 we derive an expression for the joint density of the number of claims until ruin and the time to ruin the special case where the claim sizes are exponentially distributed is also considered in which the inter arrival times are not necessarily infinitely divisible finally examples of the erlang n risk model with exponential distributed claims and the gamma risk model with erlang 2 distributed claims are given respectively to illustrate the results

solutions of the strong hamburger moment problem abstract the strong hamburger moment problem for a bi infinite sequence c n n 0 1 2 can be described as follows 1 find conditions for the existence of a positive measure on such that c n t n d t for all n 2 when there is a solution find conditions for uniqueness of the solution 3 when there is more than one solution describe the family of all solutions in this paper a theory concerning question 3 is developed in particular an analog to the nevanlinna parametrization describing the solutions of the classical hamburger moment problem is given

orthogonality and recurrence for ordered laurent polynomial sequences we consider orderings of nested subspaces of the space of laurent polynomials on the real line more general than the balanced orderings associated with the ordered bases 1 z 1 z z 2 z 2 and 1 z z 1 z 2 z 2 we show that with such orderings the sequence of orthonormal laurent polynomials determined by a positive linear functional satisfies a three term recurrence relation reciprocally we show that with such orderings a sequence of laurent polynomials which satisfies a recurrence relation of this form is orthonormal with respect to a certain positive functional

adaptive fuzzy decentralized output feedback control for nonlinear large scale systems with unknown dead zone inputs in this paper the problem of adaptive fuzzy decentralized backstepping control is considered for a class of nonlinear large scale strict feedback systems with unknown dead zones and immeasurable states fuzzy logic systems are used to approximate the unknown nonlinear functions and a state filter is designed to estimate the immeasurable states applying an adaptive backstepping design technique and combining it with the dead zone inverse method an adaptive fuzzy decentralized output feedback backstepping control is developed it is proved that all the signals of the resulting closed loop adaptive control system are semiglobally uniformly ultimately bounded and tracking errors converge to a small neighborhood of the origin by appropriate choice of design parameters simulation results are given to demonstrate that the proposed adaptive decentralized control approach has a satisfactory control performance

radau and lobatto type quadratures associated with strong stieltjes distributions in this paper n point quadrature rules associated with a strong distribution on a b 0 a b and exactly integrating certain subspaces of laurent polynomials with the highest possible dimension are first analyzed here the basic ingredients are the theory of orthogonal laurent polynomials for the distribution and an arbitrary ordering accordingly established when considering the case of quadrature rules with one node radau or two nodes lobatto previously fixed in a b the properties of the so called pseudo and quasi orthogonal laurent polynomials will play a fundamental role

matrix methods for quadrature formulas on the unit circle a survey in this paper we give a survey of some results concerning the computation of quadrature formulas on the unit circle like nodes and weights of gauss quadrature formulas for the estimation of integrals with respect to measures on the real line can be computed from the eigenvalue decomposition of the jacobi matrix szeg quadrature formulas for the approximation of integrals with respect to measures on the unit circle can be obtained from certain unitary five diagonal or unitary hessenberg matrices that characterize the recurrence for an orthogonal laurent polynomial basis these quadratures are exact in a maximal space of laurent polynomials orthogonal polynomials are a particular case of orthogonal rational functions with prescribed poles more general szeg quadrature formulas can be obtained that are exact in certain spaces of rational functions in this context the nodes and the weights of these rules are computed from the eigenvalue decomposition of an operator mobius transform of the same five diagonal or hessenberg matrices

the strong hamburger moment problem and self adjoint operators in hilbert space the strong hamburger moment problem is solved using spectral theory of unbounded self adjoint linear operators in hilbert space by means of orthogonal projections of finite dimensional range certain two point pade approximants to the moment generating function are obtained

orthogonal laurent polynomials and the strong hamburger moment problem abstract this paper is concerned with the strong hamburger moment problem shmp for a given double sequence of real numbers c c n does there exist a real valued bounded non decreasing function on with infinitely many points of increase such that for every integer n c n t n d t necessary and sufficient conditions for the existence of such a function are given in terms of the positivity of certain hankel determinants associated with c our approach is made through the study of orthogonal and quasi orthogonal laurent polynomials referred to here as l polynomials and closely related gaussian type quadrature formulas in the proof of sufficiency an inner product for l polynomials is defined in terms of the given double sequence c since orthogonal l polynomials are believed to be of interest in themselves some examples of specific systems are considered

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

nevanlinna matrices for the strong stieltjes moment problem abstract let c n n be a doubly infinite sequence of real numbers a solution of the strong hamburger moment problem is a positive measure on such that c n u n d u for n 0 1 2 a solution of the strong stieltjes moment problem is a positive measure on 0 such that c n 0 u n d u for n 0 1 2 a moment problem is indeterminate if there exists more than one solution with an indeterminate strong hamburger moment problem there is associated a nevanlinna matrix of functions holomorphic in ofc 0 these functions have growth properties partly similar to properties of analogous entire functions associated with an indeterminate classical hamburger moment problem in this paper we obtain a stronger growth result in the case where the strong stieltjes moment problem is solvable

holomorphic functions associated with indeterminate rational moment problems we consider indeterminate rational moment problems on the real line with their associated orthogonal rational functions there exists a nevanlinna type parametrization relating to the problem with associated nevanlinna matrices of functions having singularities in the closure of the set of poles of the rational functions belonging to the problem we prove that the growth at the essential singularities of the four functions in the nevanlinna matrix is the same

quadratures associated with pseudo orthogonal rational functions on the real half line with poles in 0 we consider a positive measure on 0 and a sequence of nested spaces 0 1 2 of rational functions with prescribed poles in 0 let f k k 0 with f 0 0 and f k k k 1 k 1 2 be the associated sequence of orthogonal rational functions the zeros of f n can be used as the nodes of a rational gauss quadrature formula that is exact for all functions in n n 1 a space of dimension 2n quasi and pseudo orthogonal functions are functions in n that are orthogonal to some subspace of n 1 both of them are generated from f n and f n 1 and depend on a real parameter t their zeros can be used as the nodes of a rational gauss radau quadrature formula where one node is fixed in advance and the others are chosen to maximize the subspace of n n 1 where the quadrature is exact the parameter t is used to fix a node at a preassigned point the space where the quadratures are exact has dimension 2n 1 in both cases but it is in n 1 n 1 in the quasi orthogonal case and it is in n n 2 in the pseudo orthogonal case

unique solvability of an extended hamburger moment problem abstract r functions are rational functions with no poles in the extended complex plane outside a given set a1 ap of points on the real axis methods from the theory of orthogonal polynomials can be extended to r functions by this means the author solved an extended hamburger moment problem given sequences of real numbers cn i n 1 2 i 1 p find conditions for a distribution function to exist such that d t 1 d t t a i m c m i m 1 2 i 1 p in this paper these methods are extended to treat conditions for the moment problem to have a unique solution the results are related to the classical limit point limit circle situation

robust control for a class of uncertain nonlinear systems adaptive fuzzy approach based on backstepping a robust adaptive fuzzy control design approach is developed for a class of multivariable nonlinear systems with modeling uncertainties and external disturbances the controller design for the overall systems has been carried out through a number of simpler controller designs for a series of the relevant auxiliary systems based on the backstepping design technique for each auxiliary system an adaptive fuzzy logic system is introduced to learn the behavior of unknown dynamics and then a robust control algorithm is employed to efficiently compensate the approximation error and the external disturbances as well it is shown that the resulting closed loop systems guarantee a satisfactory transient and asymptotic performance a simulation example is also presented to illustrate the design procedure and controller performance

para orthogonal laurent polynomials and the strong stieltjes moment problem abstract on the space a of laurent polynomials we consider a linear functional l which is positive definite on 0 and is defined in terms of a given bisequence c k k for each 0 we define a sequence n n z n 0 of rational functions in terms of two sequences of orthogonal laurent polynomials q n z n 0 and q n z n 0 which span a in the order 1 z 1 z z 2 z 2 and 1 z z 1 z 2 z 2 respectively it is shown that the numerators and denominators of each n n z are linear combinations of the canonical numerators and denominators of a modified pc fraction consequently n 2 n z n 0 and n 2 n 1 z n 0 converge uniformly on compact subsets of c 0 to analytic functions and hence lead to additional solutions to the strong stieltjes moment problem

the use of rational functions in numerical quadrature quadrature problems involving functions that have poles outside the interval of integration can profitably be solved by methods that are exact not only for polynomials of appropriate degree but also for rational functions having the same or the most important poles as the function to be integrated constructive and computational tools for accomplishing this are described and illustrated in a number of quadrature contexts the superiority of such rational polynomial methods is shown by an analysis of the remainder term and documented by numerical examples

fuzzy approximation based adaptive control of strict feedback nonlinear systems with time delays this paper focuses on the problem of adaptive control for a class of nonlinear time delay systems with unknown nonlinearities and strict feedback structure based on the lyapunov krasovskii functional approach a state feedback adaptive controller is constructed by backstepping the proposed adaptive controller guarantees that the system output converges into a small neighborhood of the reference signal and all the signals of the closed loop system remain bounded compared with the results that exist the main advantage of the proposed method is that the controller design is independent of the choice of the fuzzy membership functions therefore a priori knowledge of fuzzy approximators is not necessary for control design and the proposed approach requires only one adaptive law for an nth order system two numerical examples are used to illustrate the effectiveness of the proposed approach

brief paper adaptive tracking of nonlinear systems with non symmetric dead zone input quite successfully adaptive control strategies have been applied to uncertain dynamical systems subject to dead zone nonlinearities however adaptive tracking of systems with non symmetric dead zone characteristics has not been fully discussed with minimal knowledge of the dead zone parameters it is shown that the controlled system preceded by a non symmetric dead zone input can be represented as an uncertain nonlinear system subject to a linear input with time varying input coefficient to cope with this problem a new adaptive compensation algorithm is employed without constructing the dead zone inverse the proposed adaptive scheme requires only the information of bounds of the dead zone slopes and treats the time varying input coefficient as a system uncertainty the new control scheme ensures bounded error trajectory tracking and assures the boundedness of all the signals in the adaptive closed loop by appropriate selections of the controller parameters we show that the smoothness of the controller does not affect the accuracy of trajectory tracking control a numerical example is included to show the effectiveness of the theoretical results

determinacy of a rational moment problem we discuss a moment problem of stieltjes type that is related to the theory of orthogonal rational functions we obtain results which lead to a sufficient condition of carleman type for determinacy of the moment problem