quadratures and orthogonality associated with the cayley transform abstract in this paper we are dealing with the approximate calculation of weighted integrals over the whole real line the method is based in passing to the unit circle by means of the so called cayley transform z i x i x and then making use of a szeg or interpolatory type quadrature formula on the unit circle in order to obtain a gauss like quadrature rule on the real line some properties concerning orthogonality maximal domains of validity of the quadratures and connections with certain orthogonal rational functions are presented finally some numerical experiments are also carried out

domain of validity of szego quadrature formulas as is well known the n point szego quadrature formula integrates correctly any laurent polynomial in the subspace span 1 z n 1 1 z 1 z z n 1 in this paper we enlarge this subspace we prove that a set of 2n linearly independent laurent polynomials are integrated correctly the obtained result is used for the construction of szego quadrature formulas illustrative examples are given

product rules on the unit circle with uniformly distributed nodes error bounds for analytic functions this paper deals with the numerical calculation of integrals over the unit circle in the complex plane by means of quadrature formulas with uniformly distributed nodes on such region several properties of the coefficients as well as error bounds for analytic functions are established

on the local theory of regular analytic matrix functions abstract a concise exposition is given of the local spectral theory of regular analytic matrix functions without any reference to elements from the global theory

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

on the structure of equations integrable by the arbitrary order linear spectral problem the general form of partial differential equations integrable by the arbitrary order linear spectral problem is found the groups of backlund transformations corresponding to these equations are constructed it is shown that partial differential equations of the class under study are hamiltonian ones some reductions of general equations are considered in particular the hamiltonian structure of the generalisations of the sine gordon equation to the groups gl n su n and so n at arbitrary n is proved

integrable equations in 2 1 dimensions associated with symmetric and homogeneous spaces generalizations of the n wave davey stewartson and kadomtsev petviashvili equations associated with homogeneous and symmetric spaces are presented these equations are 2 1 dimensional generalizations of those presented by fordy and kulish commun math phys 89 427 1983 and athorne and fordy j phys a 20 1377 1987 examples are explicitly presented that are associated with the simplest spaces in particular a new single component 2 1 dimensional generalization of the kdv equation is presented

inversion of operator pencils on banach space using jordan chains when the generalized resolvent has an isolated essential singularity abstract we assume that the generalized resolvent for a bounded linear operator pencil mapping one banach space onto another has an isolated essential singularity at the origin and is analytic on some annular region of the complex plane centred at the origin in such cases the resolvent operator can be represented on the annulus by a convergent laurent series and the spectral set has two components a bounded component inside the inner boundary of the annulus and an unbounded component outside the outer boundary in this paper we prove that the complementary spectral separation projections on the domain space are uniquely determined by the respective generating subspaces for the associated infinite length generalized jordan chains of vectors and that the domain space is the direct sum of these two subspaces we show that the images of the generating subspaces under the mapping defined by the pencil provide a corresponding direct sum decomposition for the range space and that this is simply the decomposition defined by the complementary spectral separation projections on the range space if the domain space has a schauder basis we show that the separated systems of fundamental equations are reduced to two semi infinite systems of matrix equations which can be solved recursively to obtain a basic solution and thereby determine the laurent series coefficients for the resolvent operator on the given annular region

the polynomial spectral problem of arbitrary order a general form of the integrable equations and backlund transformations the generalisation of the akns method ablowitz et al 1974 to the matrix polynomial spectral problem of arbitrary order is given both the general form of the integrable partial differential equations and their backlund transformations are described

quadrature formulas associated with rogers szegő polynomials this paper is concerned with the estimation of integrals of email protected functions with respect to the hermite weight function by passing to the unit circle of the complex plane and considering szego and interpolatory type quadrature formulas with respect now to the rogers szego weight function several numerical experiments are given along with estimations of the error

multiple orthogonal polynomials on the unit circle normality and recurrence relations multiple orthogonal polynomials on the unit circle mopuc were introduced by j minguez and w van assche for the first time in 2008 some applications were given there and recurrence relations were obtained from a riemann hilbert problem this paper is a second contribution to this field we first obtain a determinantal formula for mopuc multiple heine s formula and we analyze the concept of normality from a dynamical point of view and by presenting a first example the combination of the lebesgue and rogers szeg measures secondly we deduce recurrence relations for mopuc without using riemann hilbert analysis only by considering orthogonality conditions this new approach allows us to complete the recurrence relations in the situation when the origin is a zero of mopuc a case that was not considered before as a consequence we give an appropriate definition of multiple verblunsky coefficients a multiple version of the well known szeg recurrence relation is also obtained here the coefficients that appear in the recurrence satisfy certain partial difference equations that are used to present a recursive algorithm for the computation of mopuc a discussion on the riemann hilbert approach that also includes the case when the origin is a zero of mopuc is presented some conclusions and open questions are finally mentioned

foreword to the proceedings of the orthoquad 2014 conference orthoquad 2014 was an international symposium on orthogonality quadrature and related topics held on january 20 24 2014 in puerto de la cruz tenerife spain it was held in memory of professor pablo gonzalez vera 1955 2012

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

the laurent expansion of pencils that are singular at the origin abstract this paper gives two new proofs of a theorem of langenhop on the laurent expansion of a matrix pencil that is singular at the origin the second based on a decomposition of van dooren leads to a computational algorithm

rational quadrature formulae on the unit circle with arbitrary poles interpolatory quadrature rules exactly integrating rational functions on the unit circle are considered the poles are prescribed under the only restriction of not lying on the unit circle a computable upper bound of the error is obtained which is valid for any choice of poles arbitrary weight functions and any degree of exactness provided that the integrand is analytic on a neighborhood of the unit circle a number of numerical examples are given which show the advantages of using such rules as well as the sharpness of the error bound also a comparison is made with other error bounds appearing in the literature

general structure of nonlinear evolution equations in 1 2 dimensions integrable by the two dimensional gelfand dickey zakharov shabat spectral problem and their transformation properties the general form of nonlinear evolution equations connected with the matrix two dimensional gelfand dickey zakharov shabat spectral problem is found the infinite dimensional abelian group of general backlund transformations and infinite dimensional abelian symmetry group for these equations are constructed

the generating operator for the n n linear system abstract the class of exactly integrable non linear evolution equations nlee related to the general first order n n linear problem is studied the set of independent scattering data j is determined and trace identities are obtained next we use a special integro differential operator and obtain the expansions of the potential and its variation in the eigenfunctions of these enable us to consider the inverse scattering method as a generalized fourier transform and to write down in a compact form the integrals of motion we prove that there exist a hierarchy of hamiltonian structures generated by at the end we calculate the symplectic form in terms of the scattering data variations

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

inversion of analytically perturbed linear operators that are singular at the origin abstract let h and k be hilbert spaces and for each z c let a z l h k be a bounded but not necessarily compact linear map with a z analytic on a region z a if a 0 is singular we find conditions under which a z 1 is well defined on some region 0 z b by a convergent laurent series with a finite order pole at the origin we show that by changing to a standard sobolev topology the method extends to closed unbounded linear operators and also that it can be used in banach spaces where complementation of certain closed subspaces is possible our method is illustrated with several key examples 2

backlund calogero group and general form of integrable equations for the two dimensional gelfand dikij zakharov shabat problem bilocal approach abstract the general form of the integrable equations and their backlund transformations connected with the general two dimensional gelfand dikij zakharov shabat spectral problem is found within the framework of the generalized akns method the bilocal tensor product of the solutions of the spectral problem is used successively which essentially simplifies the calculations of recursion operators the transformation properties of the integrable equations and backlund transformations under the gauge group are discussed

on the rogers szego polynomials an orthogonality relation on the full real line for the rogers szego polynomials is discussed it is argued that fourier transformation with the standard exponential kernel exp ixy relates the rogers szego and stieltjes wigert functions

quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity in this paper we investigate the szego radau and szego lobatto quadrature formulas on the unit circle these are n m point formulas for which m nodes are fixed in advance with m 1 and m 2 respectively and which have a maximal domain of validity in the space of laurent polynomials this means that the free parameters free nodes and positive weights are chosen such that the quadrature formula is exact for all powers z j email protected email protected p with p p n m as large as possible