schur recursions error formulas and convergence of rational estimators for stationary stochastic sequences an exact and approximate realization theory for estimation and model filters of second order stationary stochastic sequences is presented the properties of j lossless matrices as a unifying framework are used necessary and sufficient conditions for the exact realization of an estimation filter and a model filter as a submatrix of a j lossless system are deduced an extension of the so called schur algorithm yields an approximate j lossless realization based on partial past information about the process the geometric properties of such partial realizations and their convergence are studied finally connections with the nevanlinna pick problem are made and how the techniques presented constitute a generalization of many aspects of the levinson szego theory of partial realizations is shown as a consequence generalized recursive formulas for reproducing kernels and christoffel darboux formulas are obtained in this paper the scalar case is considered the matrix case will be considered in a separate publication

seeg dipole source localization based on an empirical bayesian approach taking into account forward model uncertainties electromagnetic brain source localization consists in the inversion of a forward model based on a limited number of potential measurements a wide range of methods has been developed to regularize this severely ill posed problem and to reduce the solution space imposing spatial smoothness anatomical constraint or sparsity of the activated source map this last criteria based on physiological assumptions stating that in some particular events e g epileptic spikes evoked potential few focal area of the brain are simultaneously actives has gained more and more interest bayesian approaches have the ability to provide sparse solutions under adequate parametrization and bring a convenient framework for the introduction of priors in the form of probabilistic density functions however the quality of the forward model is rarely questioned while this parameter has undoubtedly a great influence on the solution its construction suffers from numerous approximation and uncertainties even when using realistic numerical models in addition it often encodes a coarse sampling of the continuous solution space due to the computational burden its inversion implies in this work we propose an empirical bayesian approach to take into account the uncertainties of the forward model by allowing constrained variations around a prior physical model in the particular context of seeg measurements we demonstrate on simulations that the method enhance the accuracy of the source time course estimation as well as the sparsity of the resulting source map results on real signals prove the applicability of the method in real contexts

on orthogonal polynomials with respect to varying measures on the unit circle let io d i be a system of orthonormal polynomials on the unit circle with respect to d l and yi m dli be a system of orthonormal polynomials on the unit circle with respect to the varying measures d u l iw z 2 z eio where wn z is a sequence of polynomials deg w n whose zeros wn i n lie in izi 1 the asymptotic behavior of the ratio of the two systems on and outside the unit circle is obtained

combined spatial and non spatial prior for inference on mri time series when modelling fmri and other mri time series data a bayesian approach based on adaptive spatial smoothness priors is a compelling alternative to using a standard generalized linear model glm on presmoothed data another benefit of the bayesian approach is that biophysical prior information can be incorporated in a principled manner however this requirement for a fixed non spatial prior on a parameter would normally preclude using spatial regularization on that same parameter we have developed a gaussian process based prior to apply adaptive spatial regularization while still ensuring that the fixed biophysical prior is correctly applied on each voxel a parameterized covariance matrix provides separate control over the variance the diagonal elements and the between voxel correlation due to off diagonal elements analysis proceeds using evidence optimization eo with variational bayes vb updates used for some parameters the method can also be applied to non linear forward models by using a linear taylor expansion centred on the latest parameter estimates applying the method to fmri with a constrained haemodynamic response function hrf shape model shows improved fits in simulations compared to using either the non spatial or spatial smoothness prior alone we also analyse multi inversion arterial spin labelling data using a non linear perfusion model to estimate cerebral blood flow and bolus arrival time by combining both types of prior information this new prior performs consistently well across a wider range of situations than either prior alone and provides better estimates when both types of prior information are relevant

some mu ntz orthogonal systems abstract let 0 1 be a given sequence of complex numbers such that i 1 and re i j 1 0 for each i and j for muntz polynomials p x i 0npix i and q x j 0mqjx j we define an inner product p q by p q 0 1 p q x d x x 2 where p q x i 0 n j 0 m p i q j x i j we obtain the muntz polynomials qn x orthogonal with respect to this inner product and connect them with the malmquist rational functions w z v 0 n 1 z 1 v v 0 n z v orthogonal on the unit circle with respect to the inner product u v 1 2 i 1 u z v z d z z several interesting properties of polynomials qn x are given

on maximal weight solutions of two matricial moment problems in the nondegenerate case abstract we first revisit the question of existence and uniqueness of maximal weight solutions of the truncated trigonometric matrix moment problem problem ttm in the nondegenerate case the starting point of the consideration is a specialized representation of the matrix weight assigned at each point on the unit circle for the solutions of nondegenerate problem ttm our approach is essentially self contained and different from the existing methods moreover based on this approach we also derive some important weight features of the so called n extremal solutions within the whole solution set of nondegenerate problem ttm and the discrete structure of them in the second part these observations inspire us to look for the corresponding weight features of the solutions of a nondegenerate moment problem for rational matrix valued functions problem r where only the theory of orthogonal matrix polynomials on the unit circle is involved

an extension of the discontinuous galerkin method for the singular poisson equation we present a numerical method for solving a singular poisson equation which solution contains jumps and kinks due to a singular right hand side equations of this type may arise e g within the pressure computation of incompressible multiphase flows the method is an extension to the well known discontinuous galerkin dg method being able to represent the jumps and kinks with subcell accuracy in the proposed method an ansatz function which already fulfills the jump condition is subtracted from the original problem thereby reducing it to a standard poisson equation without a jump invoking a technique that we refer to as patching the construction of the ansatz function can be limited to a very narrow domain around the jump position thus making the construction numerically cheap and easy under optimal conditions the method shows a convergence order of p 1 for dg polynomial degree p still in the worst case a convergence order of approximately 2 4 is preserved for dg polynomial degree of 2

cortical surface based analysis ii inflation flattening and a surface based coordinate system the surface of the human cerebral cortex is a highly folded sheet with the majority of its surface area buried within folds as such it is a difficult domain for computational as well as visualization purposes we have therefore designed a set of procedures for modifying the representation of the cortical surface to i inflate it so that activity buried inside sulci may be visualized ii cut and flatten an entire hemisphere and iii transform a hemisphere into a simple parameterizable surface such as a sphere for the purpose of establishing a surface based coordinate system r 1999 academic press

finite surface discretization for incompressible navier stokes equations and coupled conservation laws abstract we present a new finite surface discretization fsd aiming at the incompressible navier stokes equations nse and other coupled conservation laws this discretization defines the velocities as surfaced averaged values living on the faces of the pressure volumes in which the mass is set to be conserved consequently the calculation of the mass balance on these control volumes is exact which allows more accurate information to be kept in the velocity field and produces a very accurate prediction of the pressure in the next time step the proposed discretization reduces the stencil size of the poisson equation in the projection method compared to the finite volume and finite difference discretizations due to highly accurate mass conservation the compact sixth order approximation of fsd can be used with an explicit fourth order pressure treatment this property greatly reduces the cost and complexity of the implementation we present the discrete evolution equation of the surface averaged velocities together with the enforcement of mass conservation and the solution procedure for the pressure the approximation of the nse under this new discretization uses a combination of finite difference and finite volume methods the proposed method is validated using standard laminar test cases we identify the conditions under which a fourth order pressure treatment can support the sixth order and eighth order approximations of the convection term using fourier analysis the performance of the method is evaluated on turbulent channel flows up to friction reynolds number of 950 the quantitative relationships between the accuracy of the solution and grid size are identified we present two performance indices for comparison with other methods at the error level of 1 per mille the proposed method is 28 times faster than the classic second order scheme

extended discontinuous galerkin methods for two phase flows the spatial discretization summary this work discusses a discontinuous galerkin dg discretization for two phase flows the fluid interface is represented by a level set and the dg approximation space is adapted such that jumps and kinks in pressure and velocity fields can be approximated sharply this adaption of the original dg space which can be performed on the fly for arbitrary interface shapes is referred to as extended discontinuous galerkin by combining this ansatz with a special quadrature technique one can regain spectral convergence properties for low regularity solutions which is demonstrated by numerical examples this work focuses on the aspects of spatial discretization and special emphasis is devoted on how to overcome problems related to quadrature small cut cells and condition number of linear systems temporal discretization will be discussed in future works copyright 2016 john wiley sons ltd

information geometry of mean field approximation i present a general theory of mean field approximation based on information geometry and applicable not only to boltzmann machines but also to wider classes of statistical models using perturbation expansion of the kullback divergence or plefka expansion in statistical physics a formulation of mean field approximation of general orders is derived it includes in a natural way the naive mean field approximation and is consistent with the thouless anderson palmer tap approach and the linear response theorem in statistical physics

lossless inverse scattering digital filters and estimation theory methods to construct rational solutions of the lossless inverse scattering lis problem for one port passive digital systems are described the first method is recursive and can be viewed as a generalization of the celebrated schur algorithm the second method is global and leads to a parametrization of what we call fundamental solutions from which all lis solutions may be constructed quite a few classical problems in estimation theory and network theory may be viewed as special cases of the lis problem with each fundamental solution there is a solution of a corresponding estimation problem leading to a prediction and a modeling filter for a given stochastic process

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

on the convergence of the rational interpolation approximant of carathe odory functions let zn m nm 1 with zn m 1 n 1 2 be an arbitrary sequence of complex numbers we study the convergence of the rational functionpn 1 qn which interpolates the caratheodory function f z 1 2 2 0u u z d t u eit at zn m 1 zn m nm 1 where d is a finite positive measure on the unit circle we also consider the relations between pn 1 qn and the orthogonal polynomials with respect to varying measures

bayesian analysis of the neuromagnetic inverse problem with l p norm priors abstract magnetoencephalography meg allows millisecond scale non invasive measurement of magnetic fields generated by neural currents in the brain however localization of the underlying current sources is ambiguous due to the so called inverse problem the most widely used source localization methods i e minimum norm and minimum current estimates mne and mce and equivalent current dipole ecd fitting require ad hoc determination of the cortical current distribution l 2 l 1 norm priors and point sized dipolar respectively in this article we perform a bayesian analysis of the meg inverse problem with l p norm priors for the current sources this way we circumvent the arbitrary choice between l 1 and l 2 norm prior which is instead rendered automatically based on the data by obtaining numerical samples from the joint posterior probability distribution of the source current parameters and model hyperparameters such as the l p norm order p using markov chain monte carlo mcmc methods we calculated the spatial inverse estimates as expectation values of the source current parameters integrated over the hyperparameters real meg data and simulated known source currents with realistic mri based cortical geometry and 306 channel meg sensor array were used while the proposed model is sensitive to source space discretization size and computationally rather heavy it is mathematically straightforward thus allowing incorporation of for instance a priori functional magnetic resonance imaging fmri information

on the convergence of rational functions orthogonal on the unit circle rational functions orthogonal on the unit circle with prescribed poles are studied we establish some convergence theorems for the orthogonal rational functions we generalize some of the results for szego polynomials to rational orthogonal functions

a staggered semi implicit discontinuous galerkin method for the two dimensional incompressible navier stokes equations in this paper we propose a new spatially high order accurate semi implicit discontinuous galerkin dg method for the solution of the two dimensional incompressible navier stokes equations on staggered unstructured curved meshes while the discrete pressure is defined on the primal grid the discrete velocity vector field is defined on an edge based dual grid the flexibility of high order dg methods on curved unstructured meshes allows to discretize even complex physical domains on rather coarse grids formal substitution of the discrete momentum equation into the discrete continuity equation yields one sparse linear equation system with four non zero blocks per element for only one scalar unknown namely the pressure the method is computationally efficient since the resulting system is not only very sparse but also symmetric and positive definite for appropriate boundary conditions furthermore all the volume and surface integrals needed by the scheme presented in this paper depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessor stage which leads to savings in terms of computational effort for the time evolution part in this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step the method is validated for polynomial degrees up to p 3 by solving some typical numerical test problems and comparing the numerical results with available analytical solutions or other numerical and experimental reference data

a staggered space time discontinuous galerkin method for the three dimensional incompressible navier stokes equations on unstructured tetrahedral meshes in this paper we propose a novel arbitrary high order accurate semi implicit space time discontinuous galerkin method for the solution of the three dimensional incompressible navier stokes equations on staggered unstructured curved tetrahedral meshes as is typical for space time dg schemes the discrete solution is represented in terms of space time basis functions this allows to achieve very high order of accuracy also in time which is not easy to obtain for the incompressible navier stokes equations similarly to staggered finite difference schemes in our approach the discrete pressure is defined on the primary tetrahedral grid while the discrete velocity is defined on a face based staggered dual grid while staggered meshes are state of the art in classical finite difference schemes for the incompressible navier stokes equations their use in high order dg schemes is still quite rare a very simple and efficient picard iteration is used in order to derive a space time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five point block system for the scalar pressure which is conveniently solved with a matrix free gmres algorithm from numerical experiments we find that the linear system seems to be reasonably well conditioned since all simulations shown in this paper could be run without the use of any preconditioner even up to very high polynomial degrees for a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point the resulting system is in addition symmetric and positive definite this allows us to use even faster iterative solvers like the conjugate gradient method the flexibility and accuracy of high order space time dg methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time the proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3d test problems and by comparing the obtained numerical results with available exact analytical solutions or with other numerical or experimental reference data to the knowledge of the authors this is the first time that a space time discontinuous galerkin finite element method is presented for the three dimensional incompressible navier stokes equations on staggered unstructured tetrahedral grids high order staggered space time dg scheme for the 3d incompressible navier stokes equations primal grid based on curved tetrahedra dual grid based on non standard five point hexahedra new discretization of the viscous terms on the main grid using lifting operators on the dual grid very simple space time pressure correction algorithm providing also high order in time all linear systems have maximum sparsity and use the smallest possible optimal five point stencil

szego functions and multipoint pade approximation abstract rational functions analogous to szego polynomials are introduced and their application in multipoint pade approximation is discussed

generalized szego theory in frequency analysis abstract a sequence of positive borel measures with infinite support on the unit circle is given converging in the weak star topology to a measure with support consisting of a finite number n 0 of mass points the sequences of monic rational functions orthogonal with respect to inner products determined by the measures are studied with the focus being on the asymptotic behavior of the zeros the main result is twofold for a fixed n n 0 n 0 of the zeros interesting zeros tend to the mass points of the limiting measure while the remaining n n 0 zeros uninteresting zeros may not converge at all however by considering subsequences limits may be obtained and the remaining zeros and their limits will be located in a closed disk z r

hierarchical bayesian estimation for meg inverse problem source current estimation from meg measurement is an ill posed problem that requires prior assumptions about brain activity and an efficient estimation algorithm in this article we propose a new hierarchical bayesian method introducing a hierarchical prior that can effectively incorporate both structural and functional mri data in our method the variance of the source current at each source location is considered an unknown parameter and estimated from the observed meg data and prior information by using the variational bayesian method the fmri information can be imposed as prior information on the variance distribution rather than the variance itself so that it gives a soft constraint on the variance a spatial smoothness constraint that the neural activity within a few millimeter radius tends to be similar due to the neural connections can also be implemented as a hierarchical prior the proposed method provides a unified theory to deal with the following three situations 1 meg with no other data 2 meg with structural mri data on cortical surfaces and 3 meg with both structural mri and fmri data we investigated the performance of our method and conventional linear inverse methods under these three conditions simulation results indicate that our method has better accuracy and spatial resolution than the conventional linear inverse methods under all three conditions it is also shown that accuracy of our method improves as mri and fmri information becomes available simulation results demonstrate that our method appropriately resolves the inverse problem even if fmri data convey inaccurate information while the wiener filter method is seriously deteriorated by inaccurate fmri information

a simple based discontinuous galerkin solver for steady incompressible flows in this paper we present how the well known simple algorithm can be extended to solve the steady incompressible navier stokes equations discretized by the discontinuous galerkin method the convective part is discretized by the local lax friedrichs fluxes and the viscous part by the symmetric interior penalty method within the simple algorithm the equations are solved in an iterative process the discretized equations are linearized and an equation for the pressure is derived on the discrete level the equations obtained for each velocity component and the pressure are decoupled and therefore can be solved sequentially leading to an efficient solution procedure the extension of the proposed scheme to the unsteady case is straightforward where fully implicit time schemes can be used various test cases are carried out the poiseuille flow the channel flow with constant transpiration the kovasznay flow the flow into a corner and the backward facing step flow using a mixed order formulation i e order k for the velocity and order k 1 for the pressure the scheme is numerically stable for all test cases convergence rates of k 1 and k in the l 2 norm are observed for velocity and pressure respectively a study of the convergence behavior of the simple algorithm shows that no under relaxation for the pressure is needed which is in strong contrast to the application of the simple algorithm in the context of the finite volume method or the continuous finite element method we conclude that the proposed scheme is efficient to solve the steady incompressible navier stokes equations in the context of the discontinuous galerkin method comprising hp accuracy

on the orthogonal rational functions with arbitrary poles and interpolation properties let zn mm 1n with zn m 1 n 1 2 be an arbitrary sequence of complex numbers we generalize the orthogonal rational functions with poles at we study the weak convergence and the interpolation properties of the orthogonal rational functions

orthogonal rational functions with poles on the unit circle let n be a sequence of not necessarily distinct points on the unit circle t z c z 1 set ln span 1 1 1 1 n l n 0ln where we have used the notation n nk 1 z k let m be a positive linear functional defined on the space l l with m r real for functions that are real on t define r s m r z s 1 z for r s l in particular if m is given as m r r ei d for some measure then r s r ei s ei d let the orthogonal system n be obtained from 1 n by orthogonalization three term recurrence relations quadrature formulas moment theory and interpolation properties connected with the functional m and the system n are discussed

favard theorem for reproducing kernels abstract consider for n 0 1 the nested spaces l n of rational functions of degree n at most with given poles 1 i i i 1 n let l 0 l n given a finite positive measure on the unit circle we associate with it an inner product on l by g g d suppose kn z w is the reproducing kernel for l n i e z k n z w w for all l n w then it is known that they satisfy a coupled recurrence relation in this paper we shall prove a favard type theorem which says that if you have a sequence of kernel functions kn z w which are generated by such a recurrence then there will be a measure supported on the unit circle so that kn is the reproducing kernel for ln the measure is unique under certain extra conditions on the points i

bayesian model averaging in eeg meg imaging in this paper the bayesian theory is used to formulate the inverse problem ip of the eeg meg this formulation offers a comparison framework for the wide range of inverse methods available and allows us to address the problem of model uncertainty that arises when dealing with different solutions for a single data in this case each model is defined by the set of assumptions of the inverse method used as well as by the functional dependence between the data and the primary current density pcd inside the brain the key point is that the bayesian theory not only provides for posterior estimates of the parameters of interest the pcd for a given model but also gives the possibility of finding posterior expected utilities unconditional on the models assumed in the present work this is achieved by considering a third level of inference that has been systematically omitted by previous bayesian formulations of the ip this level is known as bayesian model averaging bma the new approach is illustrated in the case of considering different anatomical constraints for solving the ip of the eeg in the frequency domain this methodology allows us to address two of the main problems that affect linear inverse solutions lis a the existence of ghost sources and b the tendency to underestimate deep activity both simulated and real experimental data are used to demonstrate the capabilities of the bma approach and some of the results are compared with the solutions obtained using the popular low resolution electromagnetic tomography loreta and its anatomically constraint version cloreta

on a class of extremal solutions of a moment problem for rational matrix valued functions in the nondegenerate case ii the main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case we will point out that each member of this family is extremal in several directions thereby the investigations below continue the studies in fritzsche et al in press 1 in doing so an application of the theory of orthogonal rational matrix functions with respect to a nonnegative hermitian matrix borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question in particular we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions we will also show that the associated riesz herglotz transform of such a particular solution admits specific representations where orthogonal rational matrix functions are involved

anatomically informed basis functions for eeg source localization combining functional and anatomical constraints distributed linear solutions have frequently been used to solve the source localization problem in eeg here we introduce an approach based on the weighted minimum norm wmn method that imposes constraints using anatomical and physiological information derived from other imaging modalities the anatomical constraints are used to reduce the solution space a priori by modeling the spatial source distribution with a set of basis functions these spatial basis functions are chosen in a principled way using information theory the reduced problem is then solved with a classical wmn method further functional constraints can be introduced in the weighting of the solution using fmri brain responses to augment spatial priors we used simulated data to explore the behavior of the approach over a range of the model s hyperparameters to assess the construct validity of our method we compared it with two established approaches to the source localization problem a simple weighted minimum norm and a maximum smoothness loreta like solution this involved simulations using single and multiple sources that were analyzed under different levels of confidence in the priors

variational inference a review for statisticians abstractone of the core problems of modern statistics is to approximate difficult to compute probability densities this problem is especially important in bayesian statistics which frames all inference about unknown quantities as a calculation involving the posterior density in this article we review variational inference vi a method from machine learning that approximates probability densities through optimization vi has been used in many applications and tends to be faster than classical methods such as markov chain monte carlo sampling the idea behind vi is to first posit a family of densities and then to find a member of that family which is close to the target density closeness is measured by kullback leibler divergence we review the ideas behind mean field variational inference discuss the special case of vi applied to exponential family models present a full example with a bayesian mixture of gaussians and derive a variant that uses stochastic optimization to scale up to massive data

solving the eeg inverse problem based on space time frequency structured sparsity constraints abstract we introduce stout spatio temporal unifying tomography a novel method for the source analysis of electroencephalograpic eeg recordings which is based on a physiologically motivated source representation our method assumes that only a small number of brain sources are active throughout a measurement where each of the sources exhibits focal smooth but localized characteristics in space time and frequency this structure is enforced through an expansion of the source current density into appropriate spatio temporal basis functions in combination with sparsity constraints this approach combines the main strengths of two existing methods namely sparse basis field expansions haufe et al 2011 and time frequency mixed norm estimates gramfort et al 2013 by adjusting the ratio between two regularization terms stout is capable of trading temporal for spatial reconstruction accuracy and vice versa depending on the requirements of specific analyses and the provided data due to allowing for non stationary source activations stout is particularly suited for the localization of event related potentials erp and other evoked brain activity we demonstrate its performance on simulated erp data for varying signal to noise ratios and numbers of active sources our analysis of the generators of visual and auditory evoked n200 potentials reveals that the most active sources originate in the temporal and occipital lobes in line with the literature on sensory processing

convergence of orthogonal rational functions outside of the unit circle rational functions orthogonal on the unit circle with prescribed poles lying outside the unit circle are studied we study the asymptotic behaviors for the orthogonal rational functions outside of the unit circle

hierarchical bayesian estimates of distributed meg sources theoretical aspects and comparison of variational and mcmc methods magnetoencephalography meg provides millisecond scale temporal resolution for noninvasive mapping of human brain functions but the problem of reconstructing the underlying source currents from the extracranial data has no unique solution several distributed source estimation methods based on different prior assumptions have been suggested for the resolution of this inverse problem recently a hierarchical bayesian generalization of the traditional minimum norm estimate mne was proposed in which the variance of distributed current at each cortical location is considered as a random variable and estimated from the data using the variational bayesian vb framework here we introduce an alternative scheme for performing bayesian inference in the context of this hierarchical model by using markov chain monte carlo mcmc strategies in principle the mcmc method is capable of numerically representing the true posterior distribution of the currents whereas the vb approach is inherently approximative we point out some potential problems related to hyperprior selection in the previous work and study some possible solutions a hyperprior sensitivity analysis is then performed and the structure of the posterior distribution as revealed by the mcmc method is investigated we show that the structure of the true posterior is rather complex with multiple modes corresponding to different possible solutions to the source reconstruction problem we compare the results from the vb algorithm to those obtained from the mcmc simulation under different hyperparameter settings the difficulties in using a unimodal variational distribution as a proxy for a truly multimodal distribution are also discussed simulated meg data with realistic sensor and source geometries are used in performing the analyses

comparing dynamic causal models this article describes the use of bayes factors for comparing dynamic causal models dcms dcms are used to make inferences about effective connectivity from functional magnetic resonance imaging fmri data these inferences however are contingent upon assumptions about model structure that is the connectivity pattern between the regions included in the model given the current lack of detailed knowledge on anatomical connectivity in the human brain there are often considerable degrees of freedom when defining the connectional structure of dcms in addition many plausible scientific hypotheses may exist about which connections are changed by experimental manipulation and a formal procedure for directly comparing these competing hypotheses is highly desirable in this article we show how bayes factors can be used to guide choices about model structure both concerning the intrinsic connectivity pattern and the contextual modulation of individual connections the combined use of bayes factors and dcm thus allows one to evaluate competing scientific theories about the architecture of large scale neural networks and the neuronal interactions that mediate perception and cognition

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

dynamical causal modelling for m eeg spatial and temporal symmetry constraints we describe the use of spatial and temporal constraints in dynamic causal modelling dcm of magneto and electroencephalography m eeg data dcm for m eeg is based on a spatiotemporal generative model of electromagnetic brain activity the temporal dynamics are described by neural mass models of equivalent current dipole ecd sources and their spatial expression is modelled by parameterized lead field functions often in classical ecd models symmetry constraints are used to model homologous pairs of dipoles in both hemispheres these constraints are motivated by assumptions about symmetric activation of bilateral sensory sources in classical approaches these constraints are hard i e the parameters of homologous dipoles are shared here in the context of dcm we illustrate the use of informed bayesian priors to implement soft symmetry constraints that are expressed in the posterior estimates only when supported by the data critically with dcm one can deploy symmetry constraints in either the temporal or spatial components of the model this enables one to test for symmetry in temporal neural mass parameters in the presence of non symmetric spatial expressions of homologous sources and vice versa furthermore we demonstrate that bayesian model comparison can be used to identify the best models among a range of symmetric and non symmetric variants our main finding is that the use of soft symmetry priors is recommended for evoked responses to bilateral sensory input we illustrate the use of symmetry constraints in dcm on synthetic and real eeg data

rates of convergence of multipoint rational approximants and quadrature formulas on the unit circle abstract in this paper multipoint rational approximants to the riesz herglotz transform of a borel measure supported on are considered we give estimates for the rate of convergence of these approximants these estimates are obtained from the asymptotic behaviour of a sequence of rational functions orthogonal with respect to and having prescribed poles 1 k where k 1 is a sequence of complex numbers compactly contained in the open unit disk as an application we also give the rate of convergence of rational szego quadrature formulas for integrals with respect to

solving the meg inverse problem a robust two way regularization method magnetoencephalography meg is a common noninvasive imaging modality for instantly measuring whole brain activities one challenge in meg data analysis is how to minimize the impact of the outliers that commonly exist in the images in this article we propose a robust two way regularization approach to solve the important meg inverse problem that is reconstructing neuronal activities using the measured meg signals the proposed method is based on the distributed source model and produces a spatio temporal solution for all the dipoles simultaneously unlike the traditional methods that use the squared error loss function our proposal uses a robust loss function which improves the robustness of the results against outliers to impose desirable spatial focality and temporal smoothness we then penalize the robust loss through appropriate spatial temporal two way regularization furthermore an alternating reweighted least squares algorithm is developed to optimize the penalized model fitting criterion

semi implicit discontinuous galerkin methods for the incompressible navier stokes equations on adaptive staggered cartesian grids in this paper a new high order semi implicit discontinuous galerkin method si dg is presented for the solution of the incompressible navier stokes equations on staggered space time adaptive cartesian grids amr in two and three space dimensions the pressure is written in the form of piecewise polynomials on the main grid which is dynamically adapted within a cell by cell amr framework according to the time dependent main grid different face based spatially staggered dual grids are defined for the piece wise polynomials of the respective velocity components although the resulting adaptive staggered grids are more complex than classical uniform cartesian meshes the numerical scheme can still be written in a rather compact form thanks to the use of a tensor product formulation for the definition of the nodal basis in the d dimensional space d 2 3 all the discrete operators can be efficiently written as a combination of linear one dimensional operators acting in the d space directions separately arbitrary high order of accuracy is achieved in space while a very simple semi implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation the real advantages of the staggered grid arise in the solution of the schur complement associated with the saddle point problem of the discretized incompressible navier stokes equations i e after substituting the discrete momentum equations into the discrete continuity equation this leads to a linear system for only one unknown the scalar pressure indeed the resulting linear pressure system is shown to be symmetric and positive definite in order to avoid a quadratic stability condition for the parabolic terms given by the viscous stress tensor an implicit discretization is also used for the diffusive terms in the momentum equation a particular feature of our staggered dg approach is that the viscous stress tensor is discretized on the dual mesh this corresponds to the use of a lifting operator but on the staggered grid both linear systems for pressure and velocity are very efficiently solved by means of a classical matrix free conjugate gradient method for which fast convergence is observed moreover it should be noticed that all test cases shown in this paper have been performed without the use of any preconditioner due to the explicit discretization of the nonlinear convective terms the final algorithm is stable within the classical cfl type time step restriction for explicit dg methods based on the fluid velocity moreover by using a suitable local time stepping technique based on the ader dg framework each element can use its optimal local time step also within the amr context the new space time adaptive staggered dg scheme has been thoroughly verified for polynomial degrees up to n 9 for a large set of non trivial test problems in two and three space dimensions for which analytical numerical or experimental reference solutions exist the high order of accuracy of the scheme is shown by means of a numerical convergence table obtained by comparing the numerical solution against a smooth analytical solution to the knowledge of the authors this is the first staggered semi implicit dg scheme for the incompressible navier stokes equations on space time adaptive meshes in two and three space dimensions

discontinuous finite element methods for incompressible flows on subdomains with non matching interfaces abstract in this paper an improved inf sup condition is derived for a class of discontinuous galerkin methods for solving the steady state incompressible stokes and navier stokes equations the computational domain is subdivided into subdomains with non matching meshes at the interfaces optimal error estimates are obtained numerical experiments including two benchmark problems are presented

on the stability of projection methods for the incompressible navier stokes equations based on high order discontinuous galerkin discretizations abstract the present paper deals with the numerical solution of the incompressible navier stokes equations using high order discontinuous galerkin dg methods for discretization in space for dg methods applied to the dual splitting projection method instabilities have recently been reported that occur for small time step sizes since the critical time step size depends on the viscosity and the spatial resolution these instabilities limit the robustness of the navier stokes solver in case of complex engineering applications characterized by coarse spatial resolutions and small viscosities by means of numerical investigation we give evidence that these instabilities are related to the discontinuous galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high order accuracy with respect to the temporal discretization this new formulation is stable in the limit of small time steps for both equal order and mixed order polynomial approximations although the dual splitting scheme itself includes inf sup stabilizing contributions we demonstrate that spurious pressure oscillations appear for equal order polynomials and small time steps highlighting the necessity to consider inf sup stability explicitly

neuroelectric source imaging using 3sco a space coding algorithm based on particle swarm optimization and l0 norm constraint the electroencephalogram eeg neuroelectric sources inverse problem is usually underdetermined and lacks a unique solution which is due to both the electromagnetism helmholtz theorem and the fact that there are fewer observations than the unknown variables one potential choice to tackle this issue is to solve the underdetermined system for a sparse solution aiming to the sparse solution a novel algorithm termed 3sco solution space sparse coding optimization is presented in this paper in 3sco after the solution space is coded with some particles the particle coded space is compressed by the evolution of particle swarm optimization algorithm where an l0 constrained fitness function is introduced to guarantee the selection of a suitable sparse solution for the underdetermined system 3sco was first tested by localizing simulated eeg sources with different configurations on a realistic head model and the comparisons with minimum norm mn loreta low resolution electromagnetic tomography l1 norm solution and focuss focal underdetermined system solver confirmed that a good sparse solution for eeg source imaging could be achieved with 3sco finally 3sco was applied to localize the neuroelectric sources in a visual stimuli related experiment and the localized areas were basically consistent with those reported in previous studies

variational free energy and the laplace approximation this note derives the variational free energy under the laplace approximation with a focus on accounting for additional model complexity induced by increasing the number of model parameters this is relevant when using the free energy as an approximation to the log evidence in bayesian model averaging and selection by setting restricted maximum likelihood reml in the larger context of variational learning and expectation maximisation em we show how the reml objective function can be adjusted to provide an approximation to the log evidence for a particular model this means reml can be used for model selection specifically to select or compare models with different covariance components this is useful in the context of hierarchical models because it enables a principled selection of priors that under simple hyperpriors can be used for automatic model selection and relevance determination ard deriving the reml objective function from basic variational principles discloses the simple relationships among variational bayes em and reml furthermore we show that em is formally identical to a full variational treatment when the precisions are linear in the hyperparameters finally we also consider briefly dynamic models and how these inform the regularisation of free energy ascent schemes like em and reml

a density problem for orthogonal rational functions abstract let n n 1 be a sequence of points in the open unit disk in the complex plane and let b 0 1 and b n z k 0 n k k k z 1 k z n 1 2 k k 1 when k 0 we put l span b n n 0 1 2 and we consider the following moment problem given a positive definite hermitian inner product on l l find a nondecreasing function on or a positive borel measure on such that f g f e i g e i d for f g l we give a necessary and sufficient condition called n extremality on a solution of the moment problem in order that l is dense in l2

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

variational bayes inference of spatial mixture models for segmentation mixture models are commonly used in the statistical segmentation of images for example they can be used for the segmentation of structural medical images into different matter types or of statistical parametric maps into activating and nonactivating brain regions in functional imaging spatial mixture models have been developed to augment histogram information with spatial regularization using markov random fields mrfs in previous work an approximate model was developed to allow adaptive determination of the parameter controlling the strength of spatial regularization inference was performed using markov chain monte carlo mcmc sampling however this approach is prohibitively slow for large datasets in this work a more efficient inference approach is presented this combines a variational bayes approximation with a second order taylor expansion of the components of the posterior distribution which would otherwise be intractable to variational bayes this provides inference on fully adaptive spatial mixture models an order of magnitude faster than mcmc we examine the behavior of this approach when applied to artificial data with different spatial characteristics and to functional magnetic resonance imaging statistical parametric maps

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

a locally conservative ldg method for the incompressible navier stokes equations in this paper a new local discontinuous galerkin method for the incompressible stationary navier stokes equations is proposed and analyzed four important features render this method unique its stability its local conservativity its high order accuracy and the exact satisfaction of the incompressibility constraint although the method uses completely discontinuous approximations a globally divergence free approximate velocity in h div is obtained by simple element by element post processing optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge this procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible navier stokes equations by solving a sequence of oseen problems numerical results are shown which verify the theoretical rates of convergence they also confirm the independence of the number of fixed point iterations with respect to the discretization parameters finally they show that the method works well for a wide range of reynolds numbers

short note an explicit expression for the penalty parameter of the interior penalty method we derive an explicit expression for the penalty parameter of the interior penalty method for elliptic problems the expression yields a coercive bilinear form and is valid for general meshes comprising of geometrically nonconforming simplical elements

variational bayesian inversion of the equivalent current dipole model in eeg meg in magneto and electroencephalography m eeg spatial modelling of sensor data is necessary to make inferences about underlying brain activity most source reconstruction techniques belong to one of two approaches point source models which explain the data with a small number of equivalent current dipoles and distributed source or imaging models which use thousands of dipoles much methodological research has been devoted to developing sophisticated bayesian source imaging inversion schemes while dipoles have received less such attention dipole models have their advantages they are often appropriate summaries of evoked responses or helpful first approximations here we propose a variational bayesian algorithm that enables the fast bayesian inversion of dipole models the approach allows for specification of priors on all the model parameters the posterior distributions can be used to form bayesian confidence intervals for interesting parameters like dipole locations furthermore competing models e g models with different numbers of dipoles can be compared using their evidence or marginal likelihood using synthetic data we found the scheme provides accurate dipole localizations we illustrate the advantage of our bayesian scheme using a multi subject eeg auditory study where we compare competing models for the generation of the n100 component

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

convergence of modified approximants associated with orthogonal rational functions abstract let n be a sequence in the unit disk d z c z consisting of a finite number of points cyclically repeated and let l be the linear space generated by the functions b n z k 0 n k z k k 1 k z let n z be orthogonal rational functions obtained from the sequence bn z orthogonalization with respect to a given functional on l and let n z be the corresponding functions of the second kind with superstar transforms n z and n z respectively interpolation and convergence properties of the modified approximants r n z u n v n u n n z v n n z u n n z v n n z that satisfy u n v n are discussed

strong and weak convergence of orthogonal systems of rational functions on the unit circle abstract pan k strong and weak convergence of orthogonal systems of rational functions on the unit circle journal of computational and applied mathematics 46 1993 427 436 we study weak and strong convergence of rational functions that are orthonormal on the unit circle and have all their poles in a given sequence of points outside the unit disk the erd s measures have been characterized as well

spatiotemporal bayesian inference dipole analysis for meg neuroimaging data recently we described a bayesian inference approach to the meg eeg inverse problem that used numerical techniques to estimate the full posterior probability distributions of likely solutions upon which all inferences were based schmidt d m george j s wood c c 1999 bayesian inference applied to the electromagnetic inverse problem human brain mapping 7 195 schmidt d m george j s ranken d m wood c c 2001 spatial temporal bayesian inference for meg eeg in nenonen j ilmoniemi r j katila t eds biomag 2000 12th international conference on biomagnetism espoo norway p 671 schmidt et al 1999 focused on the analysis of data at a single point in time employing an extended region source model they subsequently extended their work to a spatiotemporal bayesian inference analysis of the full spatiotemporal meg eeg data set here we formulate spatiotemporal bayesian inference analysis using a multi dipole model of neural activity this approach is faster than the extended region model does not require use of the subject s anatomical information does not require prior determination of the number of dipoles and yields quantitative probabilistic inferences in addition we have incorporated the ability to handle much more complex and realistic estimates of the background noise which may be represented as a sum of kronecker products of temporal and spatial noise covariance components this reduces the effects of undermodeling noise in order to reduce the rigidity of the multi dipole formulation which commonly causes problems due to multiple local minima we treat the given covariance of the background as uncertain and marginalize over it in the analysis markov chain monte carlo mcmc was used to sample the many possible likely solutions the spatiotemporal bayesian dipole analysis is demonstrated using simulated and empirical whole head meg data

strong and weak convergence of orthogonal polynomials le cercle unite fonctionnelles continues sur des mesures resultats principaux sur le cercle unite corollaires de convergence faible les parties absolument continues et les parties singulieres d une mesure integrabilite uniforme et inegalite de turan series orthogonales avec trous la ligne reelle convergence forte convergence faible convergence forte pour des moyennes arithmetiques

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

asymptotic properties of zeros of orthogonal rational functions abstract a sequence x m m 0 1 2 of real numbers gives rise to an absolutely continuous measure n t on the unit circle given by d t n d 12 n 1 m 0 x m t m e im 2 where n is a natural number t 0 1 each measure n t determines a sequence n n t of monic orthogonal rational functions with prescribed poles outside the closed unit disk in particular these rational functions may be polynomials a measure 0 with support consisting of the points 1 n0 on the unit circle is given let n n0 under a suitable condition on weak convergence of the measures and with a proper ordering of the zeros zk n n t of n n t it can be shown that lim t 1 lim t z k n n t k k 1 n 0 and that there exists a positive number n z k n n t n for k n0 1 n for all n and all t 0 1 the theory is applied to the problem of determining unknown frequencies in a trigonometric signal x m ij iajeimwj m 0 1 2

a favard theorem for rational functions abstract the space r of r functions consists of all rational functions whose poles are contained in a fixed finite subset of the finite complex plane let qn be a monic regular sequence of elements in r it is shown that qn is an orthogonal sequence with respect to a linear functional on r iff the qn satisfy a certain set of three term recurrence relations

meg source localization under multiple constraints an extended bayesian framework reconstructing the sources of magnetoencephalographic meg measurements is an ill posed inverse problem which does not admit a unique solution forward modeling of the observed signal and prior information or constraints on the solution are needed many types of constraints have been considered however a general framework is required that can accommodate for multiple priors and enables the evaluation of their contribution with this in mind we developed an empirical bayesian framework based on hierarchical linear models our approach entails the estimation of both the expected source distribution and its conditional variance the latter being constrained by an empirically determined mixture of prior variance components corresponding to the constraints one wants to introduce finally a second level of inference based on bayesian model selection enables one to compare different combinations of priors we describe here the methodological framework the estimation scheme and illustrate its application on a single subject meg data set

an empirical bayesian solution to the source reconstruction problem in eeg distributed linear solutions of the eeg source localisation problem are used routinely in contrast to discrete dipole equivalent models distributed linear solutions do not assume a fixed number of active sources and rest on a discretised fully 3d representation of the electrical activity of the brain the ensuing inverse problem is underdetermined and constraints or priors are required to ensure the uniqueness of the solution in a bayesian framework the conditional expectation of the source distribution given the data is attained by carefully balancing the minimisation of the residuals induced by noise and the improbability of the estimates as determined by their priors this balance is specified by hyperparameters that control the relative importance of fitting and conforming to various constraints here we formulate the conventional weighted minimum norm wmn solution in terms of hierarchical linear models an expectation maximisation em algorithm is used to obtain a restricted maximum likelihood reml estimate of the hyperparameters before estimating the maximum a posteriori solution itself this procedure can be considered a generalisation of previous work that encompasses multiple constraints our approach was compared with the classic wmn and maximum smoothness solutions using a simplified 2d source model with synthetic noisy data the reml solution was assessed with four types of source location priors no priors accurate priors inaccurate priors and both accurate and inaccurate priors the reml approach proved useful as 1 the regularisation or influence of the a priori source covariance increased as the noise level increased 2 the localisation error le was negligible when accurate location priors were used 3 when accurate and inaccurate location priors were used simultaneously the solution was not influenced by the inaccurate priors the reml solution was then applied to real somatosensory evoked responses to illustrate the application in an empirical setting c 2004 elsevier inc all rights reserved

a high order discontinuous galerkin method for the unsteady incompressible navier stokes equations we present a high order discontinuous galerkin discretization of the unsteady incompressible navier stokes equations in convection dominated flows using triangular and tetrahedral meshes the scheme is based on a semi explicit temporal discretization with explicit treatment of the nonlinear term and implicit treatment of the stokes operator the nonlinear term is discretized in divergence form by using the local lax friedrichs fluxes thus local conservativity is inherent spatial discretization of the stokes operator has employed both equal order p k p k and mixed order p k p k 1 velocity and pressure approximations a second order approximate algebraic splitting is used to decouple the velocity and pressure calculations leading to an algebraic helmholtz equation for each component of the velocity and a consistent poisson equation for the pressure the consistent poisson operator is replaced by an equivalent in stability and convergence operator namely that arising from the interior penalty discretization of the standard poisson operator with appropriate boundary conditions this yields a simpler and more efficient method characterized by a compact stencil size we show the temporal and spatial behavior of the method by solving some popular benchmarking tests for an unsteady stokes problem second order temporal convergence is obtained while for the taylor vortex test problem on both semi structured and fully unstructured triangular meshes spectral convergence with respect to the polynomial degree k is obtained by studying the orr sommerfeld stability problem we demonstrate that the p k p k method yields a stable solution while the p k p k 1 formulation leads to unphysical instability the good performance of the method is further shown by simulating vortex shedding in flow past a square cylinder we conclude that the proposed discontinuous galerkin method with the p k p k formulation is an efficient scheme for accurate and stable solution of the unsteady navier stokes equations in convection dominated flows

extensions of szego s theory of rational functions orthogonal on the unit circle rational functions orthogonal on the unit circle with prescribed poles lying outside the unit circle are studied we study the asymptotic behaviors for the orthogonal rational functions when the measure does not satisfy the szego condition

a high order semi explicit discontinuous galerkin solver for 3d incompressible flow with application to dns and les of turbulent channel flow abstract we present an efficient discontinuous galerkin scheme for simulation of the incompressible navier stokes equations including laminar and turbulent flow we consider a semi explicit high order velocity correction method for time integration as well as nodal equal order discretizations for velocity and pressure the non linear convective term is treated explicitly while a linear system is solved for the pressure poisson equation and the viscous term the key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under resolved high reynolds number flows as well as small time steps this penalty method is similar to the grad div stabilization widely used in continuous finite elements we further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations the solver is specifically designed for large scale computations through matrix free linear solvers including efficient preconditioning strategies and tensor product elements which have allowed us to scale this code up to 34 4 billion degrees of freedom and 147 456 cpu cores we validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large eddy simulation of turbulent channel flow at r e 180 as well as 590

on orthogonal systems of rational functions on the unit circle and polynomials orthogonal with respect to varying measures abstract we study the recurrence relations of rational functions that are orthonormal on the unit circle and have all their poles in a given sequence of points outside the unit disk these relations enable us to find the recurrence relations and the ratio asymptotic behaviors for the orthonormal polynomials with respect to varying weights d n and d n 1 at the end of this paper we also consider the convergence theorem for multipoint pade approximations of markov type functions