moment matrices and multi component kp with applications to random matrix theory questions on random matrices and non intersecting brownian motions have led to the study of moment matrices with regard to several weights the main result of this paper is to show that the determinants of such moment matrices satisfy upon adding one set of time deformations for each weight the multi component kp hierarchy these determinants are thus tau functions for these integrable hierarchies the tau functions so obtained with appropriate shifts of the time parameters forward and backwards will be expressed in terms of multiple orthogonal polynomials for these weights and their cauchy transforms the main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials it concerns an identity between the orthogonality of polynomials on the real line the bilinear identity in kp theory and a generating functional for the full kp theory an additional fact not discussed in this paper is that these functions satisfy virasoro constraints with respect to these time parameters

connection between orthogonal polynomials on the unit circle and bounded interval in this paper we establish the connection between measures on a bounded interval and on the unit circle by a transformation related with the szego classical transformation we transform a measure on the interval 1 1 into a measure on 0 2 in the same form in which the orthogonality measure of the chebyshev polynomials of fourth kind becomes the lebesgue measure we relate the sequences of orthogonal polynomials with respect to both measures and we also relate the coefficients of the three term recurrence relation with the schur parameters when the measures belong to the szego class we study the asymptotic behavior of the orthogonal polynomials on the interval outside the support of the measure as well as inside we also transform the generalized polynomials and we study the orthogonality properties of the new polynomials obtaining new interesting results and finally we solve two inverse problems connected with the transformation studied

the stokes phenomenon in the confluence of the hypergeometric equation using riccati equation abstract in this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one we study the consequence of the divergence of solutions at the irregular singular point for the unfolded system our study covers a full neighborhood of the origin in the confluence parameter space in particular we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system for this study we consider values of the confluence parameter taken in two sectors covering the complex plane in each sector we study the monodromy of a first integral of a riccati system related to the hypergeometric equation then on each sector we include the presence of logarithmic terms into a continuous phenomenon and view a stokes multiplier related to a 1 summable solution as the limit of an obstruction that prevents a pair of eigenvectors of the monodromy operators one at each singular point to coincide

szegő transformation and zeros of analytic perturbations of chebyshev weights abstract in this contribution we present a method to find the closest zeros to 1 for orthogonal polynomials with respect to analytic perturbations of the chebyshev weight functions in 1 1 the error order we obtain is o n 6

a christoffel darboux formula and a favard s theorem for orthogonal laurent polynomials on the unit circle let f k z k 0 be the family of orthonormal laurent polynomials on the unit circle which spans d in the ordering induced by p n e n 1 2 from the three term recurrence relation satisfied by f k z k 0 we deduce a christoffel darboux formula particular examples are considered and a favard type theorem is proved a connection with the ordering induced by p n e n 2 is also established

schur flow for orthogonal polynomials on the unit circle and its integrable discretization a one parameter deformation of the measure of orthogonality for orthogonal polynomials on the unit circle is considered the corresponding dynamics of the schur parameters of the orthogonal polynomials is shown to be characterized by the complex semi discrete modified kdv equation namely the schur flow a discrete analogue of the miura transformation is found an integrable discretization of the schur flow enables us to compute a pade approximation of the caratheodory functions or equivalently to compute a perron caratheodory continued fraction in a polynomial time

asymptotics for szego polynomials associated with wiener levinson filters abstract pan k asymptotics for szego polynomials associated with wiener levinson filters journal of computational and applied mathematics 46 1993 387 394 both weak and strong convergence theorems are given to support a jones njastad and saff conjecture that unknown frequencies in a periodic discrete time signal can be determined by zeros of szego polynomials with respect to some distribution function

szego polynomials applied to frequency analysis abstract this paper is concerned with the problem of determining unknown frequencies 1 i using an observed discrete time signal xn xn m arising from a continuous waveform that is the superposition of a finite number of sinusoidal waves with well defined frequencies j periods p j 2 n j n z j 1 i we investigate the wiener levinson method formulated here in terms of szeg polynomials n n z orthogonal on the unit circle with respect to a distribution function n defined by the n observed values of the signal we prove that if n0 denotes the number of critical points eiwj then for every n n0 and n 1 the zeros z j n n of n n z can be arranged so that limn z j n n eiwj for each of the frequencies wj this result confirms one of the main parts of the conjecture given by jones njastad and saff this journal 1990 on the convergence of zeros of the szeg polynomials a related result on the convergence of corresponding two point pade approximants is also given

orthogonal laurent polynomials on the unit circle extended cmv ordering and 2d toda type integrable hierarchies abstract we connect the theory of orthogonal laurent polynomials on the unit circle and the theory of toda like integrable systems using the gauss borel factorization of a cantero moral velazquez moment matrix that we construct in terms of a complex quasi definite measure supported on the unit circle the factorization of the moment matrix leads to orthogonal laurent polynomials on the unit circle and the corresponding second kind functions we obtain jacobi operators 5 term recursion relations christoffel darboux kernels and corresponding christoffel darboux formulas from this point of view in a completely algebraic way we generalize the cantero moral velazquez sequence of laurent monomials recursion relations christoffel darboux kernels and corresponding christoffel darboux formulas in this extended context we introduce continuous deformations of the moment matrix and we show how they induce a time dependent orthogonality problem related to a toda type integrable system which is connected with the well known toeplitz lattice we obtain the lax and zakharov shabat equations using the classical integrability theory tools we explicitly derive the dynamical system associated with the coefficients of the orthogonal laurent polynomials and we compare it with the classical toeplitz lattice dynamical system for the verblunsky coefficients of szeg polynomials for a positive measure discrete flows are introduced and related to darboux transformations finally we obtain the representation of the orthogonal laurent polynomials and their second kind functions using the formalism of miwa shifts in terms of functions and the subsequent bilinear equations

matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems matrix orthogonal laurent polynomials in the unit circle and the theory of toda like integrable systems are connected using the gauss borel factorization of two left and a right cantero morales velazquez block moment matrices which are constructed using a quasi definite matrix measure a block gauss borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal laurent polynomials and matrix szeg polynomials which can be expressed in terms of schur complements of bordered truncations of the block moment matrix the corresponding block extension of the christoffel darboux theory is derived deformations of the quasidefinite matrix measure leading to integrable systems of toda type are studied the integrable theory is given in this matrix scenario wave and adjoint wave functions lax and zakharov shabat equations bilinear equations and discrete flows connected with darboux transformations we generalize the integrable flows of the cafasso s matrix extension of the toeplitz lattice for the verblunsky coefficients of szeg polynomials an analysis of the miwa shifts allows for the finding of interesting connections between christoffel darboux kernels and miwa shifts of the matrix orthogonal laurent polynomials

complete integrability of the kadomtsev petviashvili equation abstract the total hierarchy of the kadomtsev petviashvili kp equation is transformed to the system of linear partial differential equations with constant coefficients the complete integrability of the kp equation is proved by using this linear system the existence and uniqueness theorem of the cauchy problem of the kp hierarchy is obtained

sequences of orthogonal laurent polynomials bi orthogonality and quadrature formulas on the unit circle in this paper the construction of orthogonal bases in the space of laurent polynomials on the unit circle is considered as an application a connection with the so called bi orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on laurent polynomials are studied

strong stieltjes distributions and orthogonal laurent polynomials with applications to quadratures and pade approximation starting from a strong stieltjes distribution general sequences of orthogonal laurent polynomials are introduced and sonic of their most relevant algebraic properties are studied from this perspective the connection between certain quadrature formulas associated with the distribution and two point pade approximants to the stieltjes transform of is revisited finally illustrative numerical examples are discussed

two point pade expansions for a family of analytic functions abstract each member g z of a family of analytic functions defined by stieltjes transforms is shown to be represented by a positive t fraction the approximants of which form the main diagonal in the two point pade table of g z the positive t fraction is shown to converge to g z throughout a domain d a b z z b a uniformly on compact subsets in addition truncation error bounds are given for the approximants of the continued function these bounds supplement previously known bounds and apply in part of the domain of g z not covered by other bounds the proofs of our results employ properties of orthogonal l polynomials laurent polynomials and l gaussian quadrature which are of some interest in themselves a number of examples are considered

on a family of symmetric hypergeometric functions of several variables and their euler type integral representation abstract this paper is devoted to the family g n of hypergeometric series of any finite number of variables the coefficients being the square of the multinomial coefficients l 1 l n l 1 l n where n z 1 all these series belong to the family of the general appell lauricella s series it is shown that each function g n can be expressed by an integral involving the previous one g n 1 thus this family can be represented by a multidimensional euler type integral what suggests some explicit link with the gelfand kapranov zelevinsky s theory of a hypergeometric systems or with the aomoto s theory of hypergeometric functions the quasi invariance of each function g n with regard to the action of a finite number of involutions of c n is also established finally a particular attention is reserved to the study of the functions g 2 and g 3 each of which is proved to be algebraic or to be expressed by the legendre s elliptic function of the first kind

characterizing the measures on the unit circle with exact quadrature formulas in the space of polynomials in the present paper we characterize the measures on the unit circle for which there exists a quadrature formula with a fixed number of nodes and weights and such that it exactly integrates all the polynomials with complex coefficients as an application we obtain quadrature rules for polynomial modifications of the bernstein measures on 1 1 having a fixed number of nodes and quadrature coefficients and such that they exactly integrate all the polynomials with real coefficients

optimization with stochastic dominance constraints we introduce stochastic optimization problems involving stochastic dominance constraints we develop necessary and sufficient conditions of optimality and duality theory for these models and show that the lagrange multipliers corresponding to dominance constraints are concave nondecreasing utility functions the models and results are illustrated on a portfolio optimization problem

the problem of center for resonant singular points of polynomial vector fields we generalize the notion of center to the case of a p q resonant singular point of a polynomial vector field in 2 and to some other situations resonant node saddle node non elementary singular point of vector field and resonant fixed point of one dimensional complex diffeomorphism we show some similarities and differences with the classical case in particular the analogue of bautin s theorem does not hold four small amplitude limit cycles can bifurcate from the center after quadratic perturbation of a quadratic vector field with resonant center

stochastic comparison of random vectors with a common copula we consider two random vectorsx andy such that the components of x are dominated in the convex order by the corresponding components of y we want to find conditions under which this implies that any positive linear combination of the components of x is dominated in the convex order by the same positive linear combination of the components of y this problem has a motivation in the comparison of portfolios in terms of risk the conditions for the above dominance will concern the dependence structure of the two random vectorsx andy namely the two random vectors will have a common copula and will be conditionally increasing this new concept of dependence is strictly related to the idea of conditionally increasing in sequence but in addition it is invariant under permutation we will actually prove that under the above conditions x will be dominated byy in the directionally convex order which yields as a corollary the dominance for positive linear combinations this result will be applied to a portfolio optimization problem

on the basic representation theorem for convex domination of measures a direct constructive proof is given for the basic representation theorem for convex domination of measures the proof is given in the finitistic case purely atomic measures with a finite number of atoms and a simple argument is then given to extend this result to the general case including both probability measures and finite borel measures on infinite dimensional spaces the infinite dimensional case follows quickly from the finite dimensional case with the use of the approximation property

about a system of anti periodic trigonometric functions in this paper we introduce the so called second kind trigonometric system which is a useful tool for the representation of 2 p anti periodic functions using these functions we study a sequence of trigonometric polynomials which are bi orthogonal in the szego s sense we study the usual topics in the theory of fourier series and we present a new connection with the orthogonal polynomials op on the unit circle and some useful properties like recurrence relations kernel representations and a favard s type theorem

stochastic order relations and lattices of probability measures we study various partially ordered spaces of probability measures and we determine which of them are lattices this has important consequences for optimization problems with stochastic dominance constraints in particular we show that the space of probability measures on mathbb r is a lattice under most of the known partial orders whereas the space of probability measures on mathbb r d typically is not nevertheless some subsets of this space defined by imposing strong conditions on the dependence structure of the measures are lattices

elementary toda orbits and integrable lattices we show that key features of several important integrable lattices appear naturally in a framework of the full toda flows using special symplectic leaves for these flows we construct a family of bi hamiltonian integrable lattices that interpolates between the nonrelativistic and relativistic toda lattices

vertex operator solutions to the discrete kp hierarchy vertex operators which are disguised darboux maps transform solutions of the kp equation into new ones in this paper we show that the bi infinite sequence obtained by darboux transforming an arbitrary kp solution recursively forward and backwards yields a solution to the discrete kp hierarchy the latter is a kp hierarchy where the continuous space x variable gets replaced by a discrete n variable the fact that these sequences satisfy the discrete kp hierarchy is tantamount to certain bilinear relations connecting the consecutive kp solutions in the sequence at the grassmannian level these relations are equivalent to a very simple fact which is the nesting of the associated infinite dimensional planes flag the discrete kp hierarchy can thus be viewed as a container for an entire ensemble of vertex or darboux generated kp solutions it turns out that many new and old systems lead to such discrete semi infinite solutions like sequences of soliton solutions with more and more solitons sequences of calogero moser systems having more and more particles just to mention a few examples this is developed in 3 in this paper as another example we show that the q kp hierarchy maps via a kind of fourier transform into the discrete kp hierarchy enabling us to write down a very large class of solutions to the q kp hierarchy this was also reported in a brief note 4

a strong szegő theorem for jacobi matrices we use a classical result of golinskii and ibragimov to prove an analog of the strong szeg theorem for jacobi matrices on l 2 mathbb n in particular we consider the class of jacobi matrices with conditionally summable parameter sequences and find necessary and sufficient conditions on the spectral measure such that sum k n infty b k and sum k n infty a k 2 1 lie in l 2 1 the linearly weighted l 2 space