interpolatory quadrature formulas on the unit circle for chebyshev weight functions in this paper interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands a comparison is made with szego quadrature formulas finally an application to the interval 1 1 is also carried out

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

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

a fast multiscale galerkin method for solving second order linear fredholm integro differential equation with dirichlet boundary conditions abstract in this paper a fast multiscale galerkin method is developed for solving second order linear fredholm integro differential equation with dirichlet boundary conditions the method is based on a matrix truncation strategy which leads to generating coefficient matrix rapidly we prove that the method is stable and has an optimal convergence order and nearly linear computational complexity up to a logarithmic factor numerical examples are presented to illustrate its computational efficiency approximation accuracy and theoretical results and to compare the computed results with those of the original multiscale galerkin method proposed recently by the same authors

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

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

numerical algorithm for solving two point second order periodic boundary value problems for mixed integro differential equations in this study the numerical solution of fredholm volterra integro differential equations for two point second order periodic boundary value problems is discussed in a reproducing kernel hilbert space a reproducing kernel hilbert space is constructed in which the periodic boundary conditions of the problem are satisfied the exact solution u x is represented in the form of series in the space w 2 3 in the mean time the n term approximate solution u n x is obtained and is proved to converge to the exact solution u x furthermore we present an iterative method for obtaining the solution in the space w 2 3 some examples are displayed to demonstrate the validity and applicability of the proposed method the numerical result indicates that the proposed method is straightforward to implement efficient and accurate for solving linear and nonlinear equations

rational functions associated with double infinite sequences of complex numbers abstract let k be a given double infinite sequence of complex numbers by defining a linear functional on the space of the laurent polynomials certain rational functions are first constructed and some algebraic properties studied the hermitian case i e k k k z is separately considered and it is shown how the theory of polynomials orthogonal on the unit circle can be used in order to prove geometric convergence for sequences such as these rational functions

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

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

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

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

on one approach to solve the linear boundary value problems for fredholm integro differential equations necessary and sufficient conditions for the well posedness of linear boundary value problems for fredholm integro differential equations are established algorithms for finding their solutions are offered

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

new method based on the hpm and rkhsm for solving forced duffing equations with integral boundary conditions this paper investigates the forced duffing equation with integral boundary conditions its approximate solution is developed by combining the homotopy perturbation method hpm and the reproducing kernel hilbert space method rkhsm hpm is based on the use of the traditional perturbation method and the homotopy technique the hpm can reduce nonlinear problems to some linear problems and generate a rapid convergent series solution in most cases rkhsm is also an analytical technique which can solve powerfully linear boundary value problems therefore the forced duffing equation with integral boundary conditions can be solved using advantages of these two methods two numerical examples are presented to illustrate the strength of the method

solving integro differential equations with cauchy kernel a simple method for solving the fredholm singular integro differential equations with cauchy kernel is proposed based on a new reproducing kernel space using a transformation and modifying the traditional reproducing kernel method the singular term is removed and the analytical representation of the exact solution is obtained in the form of series in the new reproducing kernel space the advantage of the approach lies in the fact that on the one hand by improving the definition of traditional inner product the representation of new reproducing kernel function becomes simple and requirement for image space of operator is weakened comparing with traditional reproducing kernel method on the other hand the approximate solution and its derivatives converge uniformly to the exact solution and its derivatives some examples are displayed to demonstrate the validity and applicability of the proposed method

ordinary representations and modular forms this paper establishes criteria for a representation gal q q gl2 qp to be modular in the sense that there is a newform f such that f fontaine and mazur have conjectured that is modular provided it is unramified outside of a finite set of primes irreducible has odd determinant and the restriction to a decomposition group dp at p satisfies certain conditions the following conjecture is a special case of fm95 ref 1 conjecture 3c

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

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

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

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

a new method for solving singular fourth order boundary value problems with mixed boundary conditions abstract in 1 2 3 4 5 6 7 8 it is very difficult to get reproducing kernel space of problem 1 this paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of fourth order in the new reproducing kernel space the numerical results are compared with both the exact solution and its n order derived functions in the example it is demonstrated that the new method is quite accurate and efficient for fourth order problems

an r t theorem for imaginary quadratic fields we prove the modularity of certain residually reducible p adic galois representations of an imaginary quadratic field assuming the uniqueness of the residual representation we obtain an r t theorem using a new commutative algebra criterion that might be of independent interest to apply the criterion one needs to show that the quotient of the universal deformation ring r by its ideal of reducibility is cyclic artinian of order no greater than the order of the congruence module t j where j is an eisenstein ideal in the local hecke algebra t the inequality is proven by applying the main conjecture of iwasawa theory for hecke characters and using a result of berger compos math 145 3 603 632 2009 this strengthens our previous result berger and klosin j inst math jussieu 8 4 669 692 2009 to include the cases of an arbitrary p adic valuation of the l value in particular cases when r is not a discrete valuation ring as a consequence we show that the eisenstein ideal is principal and that t is a complete intersection

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

a multiscale galerkin method for second order boundary value problems of fredholm integro differential equation in this paper multiscale galerkin method is presented to approximate the solutions of second order boundary value problems of fredholm integro differential equation the method is based on traditional galerkin method and uses the multiscale orthonormal bases to discretize the equations the proposed method is proved to be stable and have the optimal convergence order numerical examples are presented to confirm the theoretical results and show that the method is computationally stable valid and accurate

the exact solution of a class of volterra integral equation with weakly singular kernel abstract in this paper the weakly singular volterra integral equations with an infinite set of solutions are investigated among the set of solutions only one particular solution is smooth and all others are singular at the origin the numerical solutions of this class of equations have been a difficult topic to analyze and have received much previous investigation the aim of this paper is to present a numerical technique for giving the approximate solution to the only smooth solution based on reproducing kernel theory applying weighted integral we provide a new definition for reproducing kernel space and obtain reproducing kernel function using the good properties of reproducing kernel function the only smooth solution is exactly expressed in the form of series the n term approximate solution is obtained by truncating the series meanwhile we prove that the derivative of approximation converges to the derivative of exact solution uniformly the final numerical examples compared with other methods show that the method is efficient

new algorithm for a class of nonlinear integro differential equations in the reproducing kernel space this paper is concerned with the new algorithm for giving the approximate solution of a class of nonlinear integro differential equations in the reproducing kernel space two integro differential equations are solved using the algorithm it turns out that this algorithm is valid

periodic boundary value problems for second order impulsive integro differential equations by developing a new comparison result and using the monotone iterative technique we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for second order nonlinear impulsive integro differential equations of mixed type

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

solving singular two point boundary value problem in reproducing kernel space in this paper we present a new method for solving singular two point boundary value problem for certain ordinary differential equation having singular coefficients its exact solution is represented in the form of series in reproducing kernel space in the mean time the n term approximation u n x to the exact solution u x is obtained and is proved to converge to the exact solution some numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other

the exact solution for solving a class nonlinear operator equations in the reproducing kernel space in this paper the author will introduce the sobolev space w 1 2 w n firstly and then give the reproducing properties of the space w 1 2 w n as these results of those a class of the nonlinear operator equation n i 1 m i j 1 a i ju f is transformed to the n dimensional linear operator equation au f finally we also give the exact solution of this nonlinear operator equation

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

how to solve nonlinear operator equation a v2 cv f in this paper we discuss the problems of how to solve linear operator equation in hilbert spaces and a kind of nonlinear operator equation a v 2 cv f in reproducing kernel space w 2 1 a b for linear operator equation we obtain a criterion about the existence of solutions if it has solutions we get the analytic representation of its minimum norm solution and the structure of its solution space for nonlinear operator equation a v 2 cv f by using the method of solving linear operator equation one exact solution is given besides we obtain the representation of its solution set final examples show our methods are effective

how to solve the equation aubu cu f in this paper we first discuss the problem of how to solve a continuous linear operator equation in separable hilbert space we give the necessary and sufficient condition for the existence of solution and the method to decide the existence for solutions of the equation is also given if the solution exists we give the representation and approxima of the minimal normal solution of the equation and formula are obtained we conduct some research of how to solve the operator equation aubu cu f we give the factorization method and characteristic value method to solve such an equation

a class of three point boundary value problems for second order impulsive integro differential equations in banach spaces abstract in this paper we consider the following boundary value problems for second order three point nonlinear impulsive integro differential equation of mixed type in a real banach space e x t f t x t x t a x t b x t t j t t k x t t k i k x t k x t t k i k x t k x t k k 1 2 m x 0 x 1 x where is the zero element of e a x t 0 t g t s x s d s b x t 0 1 h t s x s d s g c d r d t s j j t s h c j j r and x t t k denotes the jump of x t at t t k x t t k denotes the jump of x t at t t k some new results are obtained for the existence and multiplicity of positive solutions of the above problems by using the fixed point index theory and fixed point theorem in the cone of strict set contraction operators meanwhile an example is worked out to demonstrate the main results

quadrature formulas based on rational interpolation we consider quadrature formulas based on interpolation using the ba sis functions 1 1 tkx k 1 2 3 on 1 1 where tk are parameters on the interval 1 1 we investigate two types of quadratures quadrature formulas of maximum accuracy which correctly integrate as many basis functions as possible gaussian quadrature and quadrature formulas whose nodes are the zeros of the orthogonal functions obtained by orthogonalizing the system of basis functions or thogonal quadrature we show that both approaches involve orthogonal polynomials with modified or varying weights which depend on the number of quadrature nodes the asymptotic distribution of the nodes is obtained as well as various interlacing properties and monotonicity results for the nodes

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

representation of exact solution for the nonlinear volterra fredholm integral equations abstract in this paper the representation of the exact solution for the nonlinear volterra fredholm integral equations will be obtained in the reproducing kernel space the exact solution is given by the form of series its approximate solution is obtained by truncating the series and a new numerical approximate method is obtained the error of the approximate solution is monotone deceasing in the sense of w 2 1 a b the intrinsic merit of the method given in the paper lies in its speedy convergence in addition it must be pointed out that additional conditions as linearization or unjustified assumptions are not required for nonlinear cases

new algorithm for second order boundary value problems of integro differential equation this paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of boundary value problems in the reproducing kernel space the analytical solution u x and approximate solution u n x are represented in terms of series for any initial function u 1 x w 2 3 0 1 we prove u n x u x u n x u x u n x u x n two numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method indicate the method is simple and effective

algebraic and spectral properties of general toeplitz matrices we consider problems related to the generalization of the classical fourier basis to a basis of rational functions with prescribed poles outside the unit disk we give some generalizations about the convergence and estimation of the fourier coefficients with respect to this generalized basis we also consider a rational generalization of the classical toeplitz matrices and consider the asymptotic distribution of their spectrum these bases and general toeplitz matrices were considered by ninness et al in the context of least squares system estimation where the prescribed poles allow to incorporate a priori knowledge into the system dynamics of the model

an improved method based on haar wavelets for numerical solution of nonlinear integral and integro differential equations of first and higher orders in this paper a novel technique is being formulated for the numerical solution of integral equations ies as well as integro differential equations ides of first and higher orders the present approach is an improved form of the haar wavelet methods aziz and siraj ul islam 2013 siraj ul islam et al 2013 the proposed modifications resulted in computational efficiency and simple applicability of the earlier methods aziz and siraj ul islam 2013 siraj ul islam et al 2013 in addition to this the new approach is being extended from ides of first order to ides of higher orders with initial and boundary conditions unlike the methods aziz and siraj ul islam 2013 siraj ul islam et al 2013 where the kernel function is being approximated by two dimensional haar wavelet the kernel function in the present case is being approximated by one dimensional haar wavelet the modified approach is easily extendable to higher order ides numerical examples are being included to show the accuracy and efficiency of the new method

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

representation of the exact solution and a stability analysis on the fredholm integral equation of the first kind in reproducing kernel space in this paper a new method is given in order to solve an ill posed problem on fredholm integral equation of the first kind the representation of the exact solution is given and the stability of the solution on fredholm integral equation of the first kind is discussed in the reproducing kernel space by the discussions a conclusion is obtained the stability problem is a well posed problem in the reproducing kernel space namely the measurement errors of the experimental data can not result in unbounded errors of the exact solution the numerical experiment shows that the new method given in the paper is valid

on the convergence of quadrature formulas connected with multipoint pade type approximation abstract let i f 1 1 f x x dx where is a complex valued integrable function we consider quadrature formulas for i f which are exact with respect to rational functions with prescribed poles contained in formula their rate of convergence is studied

on deformation rings of residually reducible galois representations and r t theorems we introduce a new method of proof for r t theorems in the residually reducible case we study the crystalline universal deformation ring r and its ideal of reducibility i of a mod p galois representation 0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non isomorphic constituents 1 and 2 under some assumptions on selmer groups associated with 1 and 2 we show that r i is cyclic and often finite using ideas and results of but somewhat different assumptions from bellaiche and chenevier we prove that i is principal for essentially self dual representations and deduce statements about the structure of r using a new commutative algebra criterion we show that given enough information on the hecke side one gets an r t theorem we then apply the technique to modularity problems for 2 dimensional representations over an imaginary quadratic field and a 4 dimensional representation over q

modularity of residual galois extensions and the eisenstein ideal for a totally real field f a finite extension f of fp and a galois character gf f unramified away from a finite set of places p p consider the bloch kato selmer group h h1 f 1 in an earlier paper of the authors it was proved that the number d of isomorphism classes of non semisimple reducible residual representations giving rise to lines in h which are modular by some f also unramified outside satisfies d n dimfh this was proved under the assumption that the order of a congruence module is greater than or equal to that of a divisible selmer group we show here that if in addition the relevant local eisenstein ideal j is non principal then d n when f q we prove the desired bounds on the congruence module and the selmer group we also formulate a congruence condition implying the non principality of j that can be checked in practice allowing us to furnish an example where d n

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

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

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

the unique solution of boundary value problems for nonlinear second order integral differential equations of mixed type in banach spaces in this paper a class of two point boundary value problems for nonlinear second order integral differential equations of mixed type is investigated in a real banach space without making any compactness type assumption we establish conditions for the existence of a unique solution of the equation and develop an iterative formula for approximation of the solution and a formula for estimating the error of the iterative solution the results we obtained generalize and improve various recent results

the exact solution of nonlinear operator equation aubu cu f abstract this paper is intended to discuss the problems of solving system of linear operator equations and a kind of nonlinear operator equation a u b u c u f for system of linear operator equations we obtain a criterion about the existence of solutions if it has solutions we get the analytic representation of its minimum norm solution and the structure of its solution space for nonlinear operator equation a u b u c u f with the help of the discussion on how to solve the system of linear operator equations the exact solution is given besides we obtain the representation of all its solutions

potential automorphy and change of weight we prove an automorphy lifting theorem for l adic representations where we impose a new condition at l which we call potentential diagonalizability this result allows for change of weight and seems to be substantially more exible than previous theorems along the same lines we derive several applications for instance we show that any irreducible totally odd essentially self dual regular weakly compatible system of l adic representations of the absolute galois group of a totally real eld is potentially automorphic and hence is pure and its l function has meromorphic continuation to the whole complex plane and satises the expected functional equation

on residually reducible representations on local rings let a be a local artinian ring with maximal ideal m and residual field k let r be an a algebra we mean by an n dimensional a representa tion of r any homomorphism of a algebra r from r to m a for such n a representation we denote by r the residual representation with values in m k it is proven by carayol that such a representation if r is absolutely n irreducible is completely determined by its trace in the reducible case one cannot generalize this theorem without assuming more hypothesis see w x the counterexample given in 1 the purpose of this note is to give a generalization in the reducible case it is the best generalization one can get see remarks

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

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