stieltjes transforms of a class of probability measures let the probability measures n n 2 3 be defined by n k n 1 n n a 0 for k n a where k n k 1 2 n are the zeros of the orthogonal polynomial pn 1 x which is obtained recursively by p0 x 0 p1 x 1 anpn 1 x an 1pn 1 x bnpn x xpn x conditions on an and bn are found such that the sequence n n 2 3 converges weakly to the dirac measure at the point zero this is achieved through the convergence of the sequence of stieltjes transforms d n t t to the function 1 typical examples are the al salam and carlitz polynomials the wall polynomials the lommel polynomials and the tricomi carlitz polynomials

sharp ill posedness and well posedness results for the kdv burgers equation the periodic case we prove that the kdv burgers is globally well posed in h 1 t with a solution map that is analytic from h 1 t to c 0 t h 1 t whereas it is ill posed in h s t as soon as s 0 small enough in view of the result of kappeler and topalov for kdv it thus appears that even if the dissipation part of the kdv burgers equation allows to lower the c infty critical index with respect to the kdv equation it does not permit to improve the c 0 critical index

the limit behavior of solutions for the cauchy problem of the complex ginzburg landau equation this paper is devoted to the study of the limit behavior as e 0 or e 0 and a 0 for the solutions of the cauchy problem of the complex ginzburg landau equation ut e u i u a i u u 0 u 0 x u0 x 4 n 4 n 2 0 0 s 0 1 and s n 2 2 moreover the sharp convergence rate in such spaces is also given 2002 john wiley sons inc

self adjointness of unbounded tridiagonal operators and spectra of their finite truncations abstract this paper addresses two different but related questions regarding an unbounded symmetric tridiagonal operator its self adjointness and the approximation of its spectrum by the eigenvalues of its finite truncations the sufficient conditions given in both cases improve and generalize previously known results it turns out that not only self adjointness helps to study limit points of eigenvalues of truncated operators but the analysis of such limit points is a key help to prove self adjointness several examples show the advantages of these new results compared with previous ones besides an application to the theory of continued fractions is pointed out

periodicity of grover walks on generalized bethe trees abstract we focus on the periodicity of the grover walk on the generalized bethe tree which is a rooted tree such that in each level the vertices have the same degree since the grover walk is induced by the underlying graph its properties depend on the graph in this paper we say that the graph induces periodic grover walks if and only if there exists k n such that the k th power of the time evolution operator becomes the identity operator our aim is to characterize such graphs we give the perfect characterizations of the generalized bethe trees which induce periodic grover walks

properties of zeros of orthogonal polynomials and related functions abstract there are many results in the literature on orthogonal polynomials concerning the way in which the zeros of a polynomial change as a parameter changes for the classical polynomials there are results due to markoff 1886 and stieltjes 1886 on the monotonic variation of zeros of jacobi and gegenbauer ultraspherical polynomials with respect to parameters our purpose here is to describe some recent results concerning formulas for the derivatives of zeros with respect to a parameter such formulas and their consequences are fairly well developed for bessel functions a limiting case of some of the classical orthogonal polynomials and have recently been explored for some orthogonal polynomials and related functions two general methods will be described the first use extensively by ismail the author and others is based on the hellmann feynman theorem which gives the derivative of an eigenvalue with respect to a parameter it can be applied in either a differential equations or a recurrence relations setting the second method developed in recent work of elbert and the author and applicable to suitable solutions of second order ordinary linear differential equations originates in work of richardson 1918 from this approach we obtain formulas for dc d where c c is a zero of the ultraspherical polynomial p n x these recover the known result that dc d 1 2 we get similar results for certain hermite functions these results are closely related to nicholson type formulas due to durand 1975 for sums of squares of solutions of second order linear ordinary differential equations nicholson type formulas are named after the corresponding formulas for bessel functions due to nicholson

cmv the unitary analogue of jacobi matrices we discuss a number of properties of cmv matrices by which we mean the class of unitary matrices studied recently by cantero moral and velazquez we argue that they play an equivalent role among unitary matrices to that of jacobi matrices among all hermitian matrices in particular we describe the analogues of well known properties of jacobi matrices foliation by co adjoint orbits a natural symplectic structure algorithmic reduction to this shape lax representation for an integrable lattice system ablowitz ladik and the relation to orthogonal polynomials as offshoots of our analysis we will construct action angle variables for the finite ablowitz ladik hierarchy and describe the long time behavior of this system 2006 wiley periodicals inc

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

on the recovery and resolution of exponential relaxation rates from experimental data iii the effect of sampling and truncation of data on the laplace transform inversion the method of singular function expansions which in previous papers in this series was used for the inversion of the laplace transform for the cases respectively of continuous data and continuous solution and discrete data and discrete solution is extended to cover the case of discrete data and continuous solution two discrete data points distributions are considered uniform and geometric for both we prove that the singular values and the singular functions in the solution space of the problem with discrete data and continuous solution converge to the singular values and singular functions of the problem with continuous data and continuous solution when the number of points tends to infinity and the distance between adjacents points tends to zero furthermore we show by means of numerical computations that for geometrically sampled data it is possible to obtain even better approximations of the greatest singular values than in the discrete to discrete case using a similar number of data points excellent approximations of the continuous to continuous case singular functions are also obtained implementation of the inversion procedure gives continuous solutions with high computational efficiency

limit points of eigenvalues of truncated tridiagonal operators let t be the tridiagonal operator ten a n e n 1 a n 1 e n 1 b n e n te 1 a 1 e 2 b 1 e 1 acting on a fixed orthonormal basis e n n 1 2 of a hilbert space h let p n be the orthogonal projection on the finite dimensional space h n spanned by the elements e 1 e 2 e n and let t n be the truncated operator t n p n tp n if t has a unique self adjoint extension then the set t there exists a sequence of eigenvalues n of t n with the property n contains the spectrum t of t and examples show that in general t t for many reasons the knowledge of the equality t a t is important in this paper sufficient conditions are presented such that t t

the characteristic function for jacobi matrices with applications abstract we introduce a class of jacobi operators with discrete spectra which is characterized by a simple convergence condition with any operator j from this class we associate a characteristic function as an analytic function on a suitable domain and show that its zero set actually coincides with the set of eigenvalues of j in that domain as an application we construct several examples of jacobi matrices for which the characteristic function can be expressed in terms of special functions in more detail we study the example where the diagonal sequence of j is linear while the neighboring parallels to the diagonal are constant

a quantum dynamical approach to matrix khrushchev s formulas khrushchev s formula is the cornerstone of the so called khrushchev theory a body of results which has revolutionized the theory of orthogonal polynomials on the unit circle this formula can be understood as a factorization of the schur function for an orthogonal polynomial modification of a measure on the unit circle no such formula is known in the case of matrix valued measures this constitutes the main obstacle to generalize khrushchev theory to the matrix valued setting which we overcome in this paper it was recently discovered that orthogonal polynomials on the unit circle and their matrix valued versions play a significant role in the study of quantum walks the quantum mechanical analogue of random walks in particular schur functions turn out to be the mathematical tool which best codify the return properties of a discrete time quantum system a topic in which khrushchev s formula has profound and surprising implications we will show that this connection between schur functions and quantum walks is behind a simple proof of khrushchev s formula via quantum diagrammatic techniques for cmv matrices this does not merely give a quantum meaning to a known mathematical result since the diagrammatic proof also works for matrix valued measures actually this path counting approach is so fruitful that it provides different matrix generalizations of khrushchev s formula some of them new even in the case of scalar measures furthermore the path counting approach allows us to identify the properties of cmv matrices which are responsible for khrushchev s formula on the one hand this helps to formalize and unify the diagrammatic proofs using simple operator theory tools on the other hand this is the origin of our main result which extends khrushchev s formula beyond the cmv case as a factorization rule for schur functions related to general unitary operators 2015 wiley periodicals inc

on the recovery and resolution of exponential relaxation rates from experimental data a singular value analysis of the laplace transform inversion in the presence of noise the problem of numerical inversion of the laplace transform is considered when the inverse function is of bounded strictly positive support the recent eigenvalue analysis of mcwhirter and pike for infinite support has been generalized by numerical calculations of singular values a priori knowledge of the support is shown to lead to increased resolution in the inversion and the number of exponentials that can be recovered in given levels of noise is calculated

on the support of the measure of orthogonality of a class of orthogonal polynomials for orthogonal polynomials of the formp nq n 1 x q nq n 1 x r nq n x xq n x q 0 x 0 q 1 x 1 where p n 0 q n 0 r email protected r and lim n p n lim n q n 1 2 a general sufficient condition is found such that the support of the measure of orthogonality is the entire interval 1 1 starting from this result more general cases of orthogonal polynomials are studied as a perturbation problem the results are applied to pollaczek polynomials random walk polynomials rwp neutron transport polynomials and generalized co recursive polynomials

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

a discrete approach to monotonicity of zeros of orthogonal polynomials we study the monotonicity with respect to a parameter of zeros of orthogonal polynomials our method uses the tridiagonal jacobi matrices arising from the three term recurrence relation for the polynomials we obtain new results on monotonicity of zeros of associated laguerre al salam carlitz meixner and pollaczek polynomials we also derive inequalities for the zeros of the al salam carlitz and meixner polynomials

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

rough solutions of the fifth order kdv equations abstract we consider the cauchy problem of the fifth order equation arising from the korteweg de vries kdv hierarchy t u x 5 u c 1 x u x 2 u c 2 u x 3 u 0 x t r u 0 x u 0 x u 0 h s r we prove a priori bound of solutions for h s r with s 5 4 and the local well posedness for s 2 the method is a short time x s b space which was first developed by ionescu kenig and tataru 13 in the context of the kp i equation in addition we use a weight on localized x s b structures to reduce the contribution of high low frequency interaction where the low frequency has large modulation as an immediate result from a conservation law we obtain that the fifth order equation in the kdv hierarchy t u x 5 u 30 u 2 x u 20 x u x 2 u 10 u x 3 u 0 is globally well posed in the energy space h 2

global well posedness and inviscid limit for the korteweg de vries burgers equation abstract considering the cauchy problem for the korteweg de vries burgers equation u t u x x x x 2 u u 2 x 0 u 0 where 0 1 and u is a real valued function we show that it is globally well posed in h s s s and uniformly globally well posed in h s s 3 4 for all 0 1 moreover we prove that for any t 0 its solution converges in c 0 t h s to that of the kdv equation if tends to 0

local well posedness for dispersion generalized benjamin ono equations in sobolev spaces abstract we prove that the cauchy problem for the dispersion generalized benjamin ono equation t u x 1 x u u u x 0 u x 0 u 0 x is locally well posed in the sobolev spaces h s for s 1 if 0 1 the new ingredient is that we generalize the methods of ionescu kenig and tataru 2008 13 to approach the problem in a less perturbative way in spite of the ill posedness results of molinet saut and tzvetkov 2001 21 moreover as a bi product we prove that if 0 1 the corresponding modified equation with the nonlinearity u u u x is locally well posed in h s for s 1 2 4

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

sharp local well posedness for a fifth order shallow water wave equation abstract in this paper we prove that the already established local well posedness in the range s 5 4 of the cauchy problem with an initial h s r data for a fifth order shallow water wave equation is extendable to s 5 4 by using the f s space this is sharp in the sense that the ill posedness in the range s 5 4 of this initial value problem is already known

some new results for chain sequence polynomials we study the class of generalized orthogonal polynomial sequences pn x n 0 satisfying a recurrence relation of the type pn x x cn pn 1 x npn 2 x n 1 where n 0 and the sequence n 1 cncn 1 n 1 constitutes a chain sequence we obtain a new characterization of in terms of the moment sequence associated with an orthogonal polynomial sequence and contribute to the solution of the problem of determining a signed orthogonalizing measure for a member of c

on the recovery and resolution of exponential relaxation rates from experimental data ii the optimum choice of experimental sampling points for laplace transform inversion in a previous paper bertero boccacci pike proc r soc lond a 383 15 1982 we investigated the benefits of a priori knowledge of finite support of the solution in the inversion of the laplace transform and determined the number of exponential components that could be recovered from given data in the presence of quantified amounts of noise the support of the data was assumed to be unrestricted in the present paper these calculations are extended to cover the case of a necessarily finite number of experimental data points we consider two cases uniform distribution and geometric distribution of data points in both cases the condition number of singular value inversions is minimized with respect to the placing of the data points for fixed numbers of data samples the results allow optimum placing of the experimental data points as a function of the ratio of given upper and lower bounds of the support of the solution the number of exponentials recoverable is less than in the unrestricted case but for 8 we find that with 32 linearly spaced data points optimally placed this reduction is rather small geometrical sampling of experimental data reduces further the number of sample points required for inversion for 8 we show that with 5 geometrically spaced data points points optimally placed the ill conditioning in the restoration of 2 to 4 exponential components is smaller than with 32 linearly spaced data samples this has important implications for the design of a number of scientific experiments and instruments

absence of the point spectrum in a class of tridiagonal operators let t be the tridiagonal operator te n c ne n 1 c n 1e n 1 b ne n te 1 c 1e 2 b 1e 1 where c n and b n are real sequences with c n 0 lim n c n c lim n b n b n 1 2 and e n is an orthonormal basis in a hilbert space h the spectrum of t consists of the interval 2c b 2c b plus a point spectrum outside the interval which may be empty finite or denumerable with accumulation points the points 2c b or 2c b here sufficient conditions are given such that the point spectrum of the operator t outside the interval 2c b 2c b is empty which means that the spectrum of t is the entire interval 2c b 2c b the results are illustrated with examples

dsf chain for biorthogonal polynomials and its application to matrix eigenvalue problems abstract a discrete time integrable system called the dsf chain is derived from the compatibility condition of spectral equations for biorthogonal polynomials associated with the discrete schur flow the dsf algorithm for computing eigenvalues of a type of matrix eigenvalue problem is designed a discrete analogue of the miura transformation between the discrete schur flow and the qd algorithm is constructed two special solutions to the dsf chain related to q special functions are presented

sharp well posedness for the benjamin equation abstract having the ill posedness in the range s 3 4 of the cauchy problem for the benjamin equation with an initial h s r data we prove that the already established local well posedness in the range s 3 4 of this initial value problem is extendable to s 3 4 and also that such a well posed property is globally valid for s 3 4

perturbation of the coefficients in the recurrence relation of a class of polynomials let p n x n 0 be a system of polynomials satisfying the recurrence relation p 1 x 0 p 0 x 1 p n 1 x h n p n 1 x c n p n x xp n x where h n c n are real sequences and h n 0 n 0 1 2 the co recursive polynomials p n x n 0 satisfy the same recurrence relation except for n 1 where p 1 x x c 0 0 it is well known that the problem of d the zeros of p n x is equivalent to the problem of determining the eigenvalues of a generalized eigenvalue problem tf af where t and a are symmetric matrices in this paper the problem of determining the zeros of the co recursive polynomials is reduced to a perturbation problem of the operators t and a perturbed by perturbations of rank one a function 1 2 k is found k 1 2 n whose zeros are the zeros of p n x k are the zeros of the polynomial p n x of degree n for 0 this function unifies many results concerning interlacing between the zeros of p n x and p n x for 0 moreover we obtain from this function similar results in the unstudied case 0

uniform well posedness and inviscid limit for the benjamin ono burgers equation abstract we prove that the cauchy problem for the benjamin ono burgers equation t u e x 2 u h x 2 u u u x 0 u x 0 u 0 x is uniformly globally well posed in h s s 1 for all e 0 1 moreover we show that as e 0 the solution converges to that of benjamin ono equation in c 0 t h s s 1 for any t 0 our results give an alternative proof for the global well posedness of the bo equation in h 1 r without using gauge transform which was first obtained by tao 2004 23 and also solve the problem addressed in tao 2004 23 about the inviscid limit behavior in h 1

on the zeros of a class of polynomials defined by a three term recurrence relation abstract let p n 1 x be the polynomial which is defined recursively by p 0 x 0 p 1 x 1 and n p n 1 x n 1 p n 1 x b n p n x xd n p n x n 1 2 n where n b n d n are real sequences with n 0 for every n 1 2 n and k terms of the sequence d n n 1 0 k n are equal to zero it is proved that if b n 0 and the sequence 2 n b n b n 1 n 1 is a chain sequence then the polynomial p n 1 x is of degree n k and has real and simple zeros different from zero this result generalizes and simplifies previously known results with respect to the class of polynomials whose zeros are real and simple

a quasiseparable approach to five diagonal cmv and fiedler matrices abstract recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper some classical classes of matrices with quasiseparable structure include tridiagonal related to real orthogonal polynomials and banded matrices unitary hessenberg matrices related to szego polynomials and semiseparable matrices as well as others hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results previous work has focused on characterizing h 1 quasiseparable matrices matrices with order one quasiseparable structure that are also upper hessenberg in this paper the authors introduce the concept of a twist transformation and use such transformations to explain the relationship between h 1 quasiseparable matrices and the subclass of 1 1 quasiseparable matrices without the upper hessenberg restriction which are related to the same systems of polynomials these results generalize the discoveries of cantero fiedler kimura moral and velazquez of five diagonal matrices related to horner and szego polynomials in the context of quasiseparable matrices

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

c algebras and numerical linear algebra given a self adjoint operator a on a hilbert space suppose that one wishes to compute the spectrum of a numerically in practice these problems often arise in such a way that the matrix of a relative to a natural basis is sparse for example discretized second order differential operators can be represented by doubly infinite tridiagonal matrices in these cases it is easy and natural to compute the eigenvalues of large n n submatrices of the infinite operator matrix and to hope that if n is large enough then the resulting distribution of eigenvalues will give a good approximation to the spectrum of a numerical analysts call this the galerkin method while this hope is often realized in practice it often fails as well and it can fail in spectacular ways the sequence of eigenvalue distributions may not converge as n or they may converge to something that has little to do with the original operator a at another level even the meaning of convergence has not been made precise in general in this paper we determine the proper general setting in which one can expect convergence and we describe the asymptotic behavior of the n n eigenvalue distributions in all but the most pathological cases under appropriate hypotheses we establish a precise limit theorem which shows how the spectrum of a is recovered from the sequence of eigenvalues of the n n compressions in broader terms our results have led us to the conclusion that numerical problems involving infinite dimensional operators require a reformulation in terms ofc algebras indeed it is only when the single operator a is viewed as an element of an appropriate c algebra a that one can see the precise nature of the limit of the n n eigenvalue distributions the limit is associated with a tracial state on a normally a is highly noncommutative and in our main applications it is a simple c algebra having a unique tracial state we obtain precise asymptotic results for operators which represent discretized hamiltonians of one dimensional quantum systems with arbitrary continuous potentials

asymptotic behavior of the verblunsky coefficients for the opuc with a varying weight we present an asymptotic analysis of the verblunsky coefficients for the polynomials orthogonal on the unit circle with the varying weight e nv cos x assuming that the potential v has four bounded derivatives on 1 1 and the equilibrium measure has a one interval support we obtain the asymptotics as a solution of the system of string equations

input recovery from noisy output data using regularized inversion of the laplace transform in a dynamical system the input is to be recovered from finitely many measurements blurred by random error of the output of the system as usual the differential equation describing the system is reduced to multiplication with a polynomial after applying the laplace transform it appears that there exists a natural unbiased estimator for the laplace transform of the output from which an estimator of the input can be obtained by multiplication with the polynomial and subsequent application of a regularized inverse of the laplace transform it is possible moreover to balance the effect of this inverse so that ill posedness remains restricted to its actual source differentiation the rate of convergence of the integrated mean square error is a positive power of the number of data the order of the differential equation has an adverse effect on the rate which on the other hand increases with the smoothness of the input as usual

absence of absolutely continuous spectrum for random scattering zippers a scattering zipper is a system obtained by concatenation of scattering events with equal even number of incoming and outgoing channels the associated scattering zipper operator is the unitary analog of jacobi matrices with matrix entries for infinite identical events and independent and identically distributed random phases lyapunov exponents positivity is proved and yields absence of absolutely continuous spectrum by kotani s theory

szegő difference equations transfer matrices and orthogonal polynomials on the unit circle we develop the theory of orthogonal polynomials on the unit circle based on the szeg recurrence relations written in matrix form the orthogonality measure and c function arise in exactly the same way as weyl s function in the weyl approach to second order linear differential equations on the half line the main object under consideration is the transfer matrix which is a key ingredient in the modern theory of one dimensional schrodinger operators discrete and continuous and the notion of subordinacy from the gilbert pearson theory we study the relations between transfer matrices and the structure of orthogonality measures the theory is illustrated by the szeg equations with reflection coefficients having bounded variation

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

bulk edge correspondence of one dimensional quantum walks we outline a theory of symmetry protected topological phases of one dimensional quantum walks we assume spectral gaps around the symmetry distinguished points 1 and 1 in which only discrete eigenvalues are allowed the phase classification by integer or binary indices extends the classification known for translation invariant systems in terms of their band structure however our theory requires no translation invariance whatsoever and the indices we define in this general setting are invariant under arbitrary symmetric local perturbations even those that cannot be continuously contracted to the identity more precisely we define two indices for every walk characterizing the behavior far to the right and far to the left respectively their sum is a lower bound on the number of eigenstates at 1 and 1 for a translation invariant system the indices add up to zero so one of them already characterizes the phase by joining two bulk phases with different indices we get a walk in which the right and left indices no longer cancel so the theory predicts bound states at 1 or 1 this is a rigorous statement of bulk edge correspondence the results also apply to the hamiltonian case with a single gap at zero

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

orthogonal polynomials on the unit circle distribution of zeros departamento de matemritica aplicada escuela tpcnica superior de ingenieros industriales universidad de vigo apartado de correos no 62 36280 vigo spain received 30 august 1990 revised 29 january 1991 abstract marcellan f and e godoy orthogonal polynomials on the unit circle distribution of zeros journal of computational and applied mathematics 37 1991 195 208 in this paper we summarize some results concerning zeros of orthogonal polynomials with respect to an indefinite inner product we analyze the inverse problem i e a discrete representation for the functional in terms of the n th orthogonal polynomial finally some numerical examples are shown keywords toeplitz matrices orthogonal polynomials on the unit circle zeros quadrature formulas

linear regularizing algorithms for positive solutions of linear inverse problems linear regularization methods provide a simple technique for determining stable approximate solutions of linear ill posed problems such as fredholm equations of the first kind cauchy problems for elliptic equations and backward solution of forward parabolic equations in most of these problems the solution must be positive to satisfy physical plausibility in this paper we consider ill posed first kind convolution equations and related problems such as numerical differentiation radon transform and laplace transform inversion we investigate several linear regularization algorithms which provide positive approximate solutions for these problems at least in the absence of errors on the data for noisy data the solution is not necessarily positive because the appearance of negative values can then only be an effect of the noise the negative part of the solution should be negligible with a suitable choice of the regularization parameter a price to pay for ensuring positivity is always however a reduction in resolution

on the well posedness for kadomtsev petviashvili burgers i equation abstract we prove local and global well posedness in h s 0 r 2 s 1 2 for the cauchy problem associated with the kadomtsev petviashvili burgers i equation kpbi by working in bourgain s type spaces this result is almost sharp if one requires the flow map to be smooth

an alternative proof of a theorem of stieltjes and related results abstract let p n x n 1 be the orthogonal polynomials defined by a n p n 1 x a n 1 p n 1 x b n p n x xp n x p 0 x 0 p 1 x 1 where both sequences a n and b n are bounded and a n 0 assume that x is the unique up to a constant distribution function which corresponds to the measure of orthogonality of p n x and denote by s the spectrum of x alternative proofs of a theorem due to stieltjes and of a conjecture due to maki concerning the limit points of s are given a typical example to the maki s conjecture together with a general result concerning the density of the zeros of the polynomials p n x covers as a particular case a theorem of chihara which generalizes the well known theorem of blumenthal

quadratic dispersive generalized benjamin ono equation abstract this paper is concerned with quadratic dispersive generalized benjamin ono equation 1 1 1 2 we prove sharp local well posedness in h s s 1 4 with a solution map that is analytic from h s r to c 0 t h s r and the ill posed result in h s r as soon as s s in the sense that the flow map u 0 u t cannot be continuous

dispersive limit from the kawahara to the kdv equation abstract we investigate the limit behavior of the solutions to the kawahara equation u t u 3 x e u 5 x u u x 0 e 0 as e 0 in this equation the terms u 3 x and e u 5 x compete each other and cancel each other at frequencies of order 1 e this prohibits the use of a standard dispersive approach for this problem nevertheless by combining different dispersive approaches according to the space frequency ranges we succeed in proving that the solutions to this equation converge in c 0 t h 1 r towards the solutions of the kdv equation for any fixed t 0

on the analytical and numerical properties of the truncated laplace transform i the laplace transform is frequently encountered in mathematics physics engineering and other fields however the spectral properties of the laplace transform tend to complicate its numerical treatment therefore the closely related truncated laplace transforms are often used in applications we have constructed efficient algorithms for the evaluation of the singular value decomposition of truncated laplace transforms in the current paper we introduce algorithms for the evaluation of the right singular functions and singular values of truncated laplace transforms algorithms for the computation of the left singular functions will be introduced separately in an upcoming paper the resulting algorithms are applicable to all environments likely to be encountered in applications including the evaluation of singular functions corresponding to extremely small singular values e g 10 1000

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation

global well posedness and nonsqueezing property for the higher order kdv type flow abstract in this paper we prove that the periodic higher order kdv type equation t u 1 j 1 x 2 j 1 u 1 2 x u 2 0 t x r t u 0 x u 0 x u 0 h s t is globally well posed in h s for s j 2 j 3 the proof is based on i method introduced by colliander et al 4 we also prove the nonsqueezing property of the periodic higher order kdv type equation the proof relies on gromov s nonsqueezing theorem for the finite dimensional hamiltonian system and an approximation argument for the solution flow more precisely after taking the frequency truncation to the solution flow we apply the nonsqueezing theorem by using the approximation argument we extend this result to the infinite dimensional system this argument was introduced by kuksin 14 and made concretely by bourgain 2 for the 1d cubic nls flow and colliander et al 5 for the kdv flow one of our observations is that the higher order kdv type equation has the better modulation effect from the non resonant interaction than that the kdv equation has hence unlike the work of colliander et al 5 we can get the nonsqueezing property for the solution flow without the miura transform