fourth order canonical forms of singular self adjoint boundary conditions abstract canonical forms of self adjoint boundary conditions are well known in the second order sturm liouville case for both regular and singular problems these are critical for the theoretical investigation of the eigenvalues as well as their numerical computation recently canonical forms have been found for fourth order regular problems these are much more complicated than the second order ones here we find canonical forms for fourth order singular problems with one or both endpoints of the domain interval singular and with arbitrary deficiency index d in the regular fourth order case d 4 in the singular case d can assume any value between 0 and 4 and moreover depends on the nature of the singularities at the two endpoints these different values of d and their dependence on the endpoints introduces serious additional complications

a general stability theorem for linear discrete time systems the strongest sufficient conditions for constraining zeros to be on or within a circular region in the z plane in terms of weighted absolute norms are derived using elementary geometry it is shown that the weighted l 1 norm yields the strongest result

boundary values for an eigenvalue problem with a singular potential abstract the radial hamiltonian operator h d 2 dx 2 x 2 is considered on 0 while no boundary condition is needed at in order to make h self adjoint one is indeed needed at 0 this article describes in elementary terms how such conditions may be found the resulting eigenvalue problem is then discussed and its relation to previous work is explained other potentials with singularities at 0 are also mentioned

legendre polynomials legendre stirling numbers and the left definite spectral analysis of the legendre differential expression in this paper we develop the left definite spectral theory associated with the self adjoint operator a k in l2 1 1 generated from the classical second order legendre differential equation ll k y t 1 t2 y ky y t 1 1 that has the legendre polynomials pm t m 0 as eigenfunctions here k is a fixed nonnegative constant more specifically for k 0 we explicitly determine the unique left definite hilbert sobolev space wn k and its associated inner product n k for each n n moreover for each n n we determine the corresponding unique left definite self adjoint operator an k in wn k and characterize its domain in terms of another left definite space the key to determining these spaces and inner products is in finding the explicit lagrangian symmetric form of the integral composite powers of ll k in turn the key to determining these powers is a remarkable new identity involving a double sequence of numbers which we call legendre stirling numbers

symmetric positive solutions to singular system with multi point coupled boundary conditions in this paper we study the existence and multiplicity of symmetric positive solutions for a nonlinear system with multi point coupled boundary conditions the arguments are based upon a specially constructed cone and the fixed point index theorem in cones an example is then given to demonstrate the applicability of our results

orthogonal polynomial solutions of linear ordinary differential equations abstract this paper surveys the latest known results concerning the classification of differential equations of the form k 1 n a k x y k x y x having a sequence of polynomial eigenfunctions p n x n 0 that are orthogonal with respect to some real bilinear form since the publishing of the erice report in 1990 several new significant results and generalizations have been discovered

orthogonal polynomials satisfying fourth order differential equations since they are rather important and quite accessible we repeat the general theoretical facts concerning weights moments and polynomials pertaining to fourth order differential equations we then briefly discuss the squares of the differential equations of the second order giving a number of easily derived examples of fourth order problems this is followed by three new orthogonal polynomial sets satisfying fourth order differential equations but which do not satisfy second order differential equations

bound for all coefficient diameters of real schur stable interval polynomials a sharp diameter bound for all coefficient intervals of real robustly schur stable polynomials is established the results may be used in a preprocessing rejection scheme when testing for robust schur stability

matrix representations of fourth order boundary value problems with periodic boundary conditions we show that a class of regular self adjoint fourth order boundary value problems bvps is equivalent to a certain class of matrix problems conversely for any given matrix problem in this class there exist fourth order self adjoint bvps which are equivalent to the given matrix problem equivalent here means that they have exactly the same eigenvalues such an equivalence is considered with the special coupled boundary conditions i e periodic boundary conditions

self adjoint extensions for linear hamiltonian systems with two singular endpoints abstract this paper is concerned with self adjoint extensions for a linear hamiltonian system with two singular endpoints the domain of the closure of the corresponding minimal hamiltonian operator h 0 is described by properties of its elements at the endpoints of the discussed interval decompositions of the domains of the corresponding left and right maximal hamiltonian operators are provided and expressions of the defect indices of h 0 in terms of those of the left and right minimal operators are given based on them characterizations of all the self adjoint extensions for a hamiltonian system are obtained in terms of square integrable solutions as a consequence the characterizations of all the self adjoint extensions are given for systems in several special cases

on necessary conditions for real robust schur stability a coefficient diameter of real schur stable interval polynomials is bounded using meromorphic functions the resulting bound is the best possible a known bound for the ratio of the two leading coefficients is improved for large diameters

self adjoint domains symplectic geometry and limit circle solutions abstract everitt and markus characterized the domains of self adjoint operator realizations of very general even and odd order symmetric ordinary differential equations in terms of lagrangian subspaces of symplectic spaces recently for the even order case with real coefficients wang sun and zettl constructed limit circle lc solutions and hao wang sun and zettl characterized the self adjoint domains in terms of lc solutions these lc solutions are higher order analogues of the celebrated titchmarsh weyl limit circle solutions in the second order case this lc characterization has been used to obtain information about the discrete continuous and essential spectra of these operators in this paper we investigate the connection between these two very different kinds of characterizations and thus add the methods of symplectic geometry to the techniques available for the investigation of the spectrum of self adjoint operators

characterization of domains of self adjoint ordinary differential operators article history the gkn glazman krein naimark theorem characterizes all self adjoint realizations of linear symmetric formally self adjoint ordinary differential equations in terms of maximal domain functions these functions depend on the coefficients and this dependence is implicit and complicated in the regular case an explicit characterization in terms of two point boundary conditions can be given in the singular case when the deficiency index d is maximal the gkn characterization can be made more explicit by replacing the maximal domain functions by a solution basis for any real or complex value of the spectral parameter i n the much more difficult intermediate cases not all solutions contribute to the singular self adjoint conditions in 1986 sun found a representation of the self adjoint singular conditions in terms of certain solutions for nonreal values of in this paper we give a representation in terms of certain solutions for real this leads to a classification of solutions as limit point lp or limit circle lc in analogy with the celebrated weyl classification in the second order case the lc solutions contribute to the singular boundary conditions the lp solutions do not the advantage of using real is not only because it is in general easier to find explicit solutions but more importantly it yields information about the spectrum

inverse spectral problem for the laguerre differential operator abstract we construct a self adjoint operator associated with the following given set of eigenvalues an c n 1 an b n 0 this problem arises in sampling theory and deals with the construction of an interpolating basis associated with a given set of sampling points the so called two sided cardinal series in order to solve this inverse spectral problem we shall need to extend the gelfand levitan theory to operators close to the laguerre differential operator

jordan structures of upper equivalent matrices abstract jordan structures of strictly lower triangular completions of matrices over an algebraically closed field are studied we give in terms of majorization a sufficient condition for the existence of a completion which preserve the spectrum of the original matrix and has prescribed jordan structure the case of nilpotent matrix was considered previously by krupnik and leibman

left definite sturm liouville problems abstract left definite regular self adjoint sturm liouville problems with either separated or coupled boundary conditions are studied we give an elementary proof of the existence of eigenvalues for these problems for any fixed equation we establish a sequence of inequalities among the eigenvalues for different boundary conditions and estimate the range of each eigenvalue as a function on the space of boundary conditions some of our results here yield an algorithm for numerically computing the eigenvalues of a left definite problem with an arbitrary coupled boundary condition our inequalities imply that the well known asymptotic formula for the eigenvalues in the separated case also holds in the coupled case moreover we study the continuous and differentiable dependence of the eigenvalues of the general left definite problem on all the parameters in its differential equation and boundary condition

continuous spectrum and square integrable solutions of differential operators with intermediate deficiency index we explore the connection between square integrable solutions for real values of the spectral parameter and the continuous spectrum of self adjoint ordinary differential operators with arbitrary deficiency index d we show that if for all in an open interval i there are d of linearly independent square integrable solutions then for every extension of dmin the point spectrum is nowhere dense in i and there is a self adjoint extension of smin which has no continuous spectrum in i this analysis is based on our construction of limit point lp and limit circle lc solutions obtained recently in an earlier paper

boundary triples and weyl m functions for powers of the jacobi differential operator abstract the abstract theory of boundary triples is applied to the classical jacobi differential operator and its powers in order to obtain the weyl m function for several self adjoint extensions with interesting boundary conditions separated periodic and those that yield the friedrichs extension these matrix valued nevanlinna herglotz m functions are to the best knowledge of the author the first explicit examples to stem from singular higher order differential equations the creation of the boundary triples involves taking pieces determined in 26 of the principal and non principal solutions of the differential equation and putting them into the sesquilinear form to yield maps from the maximal domain to the boundary space these maps act like quasi derivatives which are usually not well defined for all functions in the maximal domain of singular expressions however well defined regularizations of quasi derivatives are produced by putting the pieces of the non principal solutions through a modified gram schmidt process

the change of the jordan structure of a mtrix under small perturbations abstract in this paper a complete description including multiplicity is given for the jordan structure of a matrix which is a small perturbation of a matrix of known jordan structure the problem solved here was solved independently as noted and the other solution has been published in english however the solutions are different the problem is important and it seems desirable to have this solution widely available

the classification of self adjoint boundary conditions separated coupled and mixed there are three basic types of self adjoint regular and singular boundary conditions separated coupled and mixed for even order problems with real coefficients one regular endpoint and arbitrary deficiency index d we give a construction for each type and determine the number of possible conditions of each type under the assumption that there are d linearly independent square integrable solutions for some real value of the spectral parameter in the separated case our construction yields non real conditions for all orders greater than two it is well known that no such conditions exist in the second order case our construction gives a direct alternative to the recent construction of everitt and markus which uses the theory of symplectic spaces we believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self adjoint boundary conditions such forms are known only in the second order i e sturm liouville case even for regular problems of order four no such forms are available

the sobolev orthogonality and spectral analysis of the laguerre polynomials l n k for positive integers k for k n we consider the analysis of the classical laguerre differential expression l k y x 1 x ke x x k 1e xy x rx ke xy x x 0 where r 0 is fixed in several nonisomorphic hilbert and hilbert sobolev spaces in one of these spaces specifically the hilbert space l2 0 x ke x it is well known that the glazman krein naimark theory produces a self adjoint operator a k generated by l k that is bounded below by rl where i is the identity operator on l2 0 x ke x consequently as a result of a general theory developed by littlejohn and wellman there is a continuum of left definite hilbert spaces h s k v s k s k s 0 and left definite self adjoint operators b s k s 0 associated with the pair l2 0 x ke x a k for a k and each of the operators b s k it is the case that the tail end sequence l n k n k of laguerre polynomials form a complete set of eigenfunctions in the corresponding hilbert spaces in 1995 kwon and littlejohn introduced a hilbert sobolev space w k 0 in which the entire sequence of laguerre polynomials is orthonormal in this paper we construct a self adjoint operator in this space generated by the second order laguerre differential expression l k having l n k n 0 as a complete set of eigenfunctions the key to this construction is in identifying a certain closed subspace of w k 0 with the kth left definite vector space v k k

jordan forms of completions of partial upper triangular matrices abstract the possible jordan forms of completions of a matrix prescribed in its upper triangular part are studied it is shown that there always exists a nonderogatory completion for such a matrix jordan forms of nilpotent completions are considered as well

on the self adjoint extensions of symmetric ordinary differential operators in direct sum spaces in this paper we give the characterizations of self adjoint domains for singular symmetric operators by describing the boundary conditions of domain of conjugate differential operator with interior singular points or in direct sum spaces last we show that there exist other self adjoint extension of t0 m which cannot be expressed as the direct sum of the self adjoint extensions over the various subintervals

singular second order operators the maximal and minimal operators and selfadjoint operators in between differential operators arising from the differential equation py qy lambda wy are put in the hilbert space setting l 2 a b w a new and simpler characterization of the minimal operator is exhibited selfadjoint operators which lie in between the minimal and maximal operator are easily described in terms of conditions on boundary coefficients which look like and indeed are the same as those imposed on regular problems examples drawn from mathematical physics include the legendre laguerre hermite and bessel problems

boundary conditions for general ordinary differential operators and their adjoints a characterisation is obtained of all the regularly solvable operators and their adjoints generated by a general differential expression in the domains of these operators are described in terms of boundary conditions involving the solutions of mu wu and the adjoint equation the results include those of sun jiong 15 concerning self adjoint realisations of a symmetric m when the minimal operator has equal deficiency indices if the deficiency indices are unequal the maximal symmetric operators are determined by the results herein another special case concerns the j self adjoint operators where j denotes complex conjugation and for this we recover the results of zai jiu shang in 16

the left definite spectral theory for the classical hermite differential equation abstract in this paper we develop the left definite spectral theory associated with the self adjoint operator a in l2 exp t2 generated from the classic second order hermite differential equation l h y t y 2ty ky y t that has the hermite polynomials hm t m 0 as eigenfunctions more specifically for each n n we explicitly determine the unique left definite hilbert sobolev space wn and associated inner product n which is generated from the nth integral power lhn of lh moreover for each n n we determine the corresponding unique left definite self adjoint operator an in wn and characterize its domain in terms of another left definite space as a consequence of this we explicitly determine the domain of each integral power of a and in particular we obtain a new characterization of the domain of the classical right definite operator a

on robust hurwitz and schur polynomials based on results of a v lipatov and n i sokolov 1979 simple sufficient conditions for the robust stability of both strictly hurwitz and schur interval polynomials are obtained furthermore sufficient conditions are derived for robustness of strictly hurwitz and schur polynomials whose coefficients of linear functions of a single parameter several examples are discussed illustrating the derived results

robust schur stability of a polytope of polynomials the main objective of the authors is to provide a necessary and sufficient condition for a polytope of polynomials to have all its zeros inside the unit circle the criterion obtained serves as a discrete time counterpart for results in s bialas 1985 and f fu and b r barmish 1987 for the continuous case also the results are reduced to operations on n 1 n 1 matrices it is concluded that by the edge result of a c bartlett et al 1987 it suffices to check the exposed edges in order to determine whether a polytope of polynomials has all its zeros in a simply connected region d

symmetrizability of differential equations having orthogonal polynomial solutions abstract we show that if a linear differential equation of spectral type with polynomial coefficients l n y x i 0 n l i x n y x has an orthogonal polynomial system of solutions then the differential operator ln must be symmetrizable we also give a few applications of this result

glazman krein naimark theory left definite theory and the square of the legendre polynomials differential operator abstract as an application of a general left definite spectral theory everitt littlejohn and wellman in 2002 developed the left definite theory associated with the classical legendre self adjoint second order differential operator a in l 2 1 1 which has the legendre polynomials p n n 0 as eigenfunctions as a consequence they explicitly determined the domain d a 2 of the self adjoint operator a 2 however this domain in their characterization does not contain boundary conditions in fact this is a general feature of the left definite approach developed by littlejohn and wellman yet the square of the second order legendre expression is in the limit 4 case at each end point x 1 in l 2 1 1 so d a 2 should exhibit four boundary conditions in this paper we show that this domain can in fact be expressed using four separated boundary conditions using the classical gkn glazman krein naimark theory in addition we determine a new characterization of d a 2 that involves four non gkn boundary conditions these new boundary conditions are surprisingly simple and natural and are equivalent to the boundary conditions obtained from the gkn theory

a spectral study of the second order exceptional x1 jacobi differential expression and a related non classical jacobi differential expression abstract the exceptional x 1 jacobi differential expression is a second order ordinary differential expression with rational coefficients it was discovered by gomez ullate kamran and milson in 2009 in their work they showed that there is a sequence of polynomial eigenfunctions p n n 1 called the exceptional x 1 jacobi polynomials there is no exceptional x 1 jacobi polynomial of degree zero these polynomials form a complete orthogonal set in the weighted hilbert space l 2 1 1 w where w is a positive rational weight function related to the classical jacobi weight among other conditions placed on the parameters and it is required that 0 in this paper we develop the spectral theory of this expression in l 2 1 1 w we also consider the spectral analysis of the extreme non exceptional case namely when 0 in this case the polynomial solutions are the non classical jacobi polynomials p n 2 n 2 we study the corresponding jacobi differential expression in several hilbert spaces including their natural l 2 setting and a certain sobolev space s where the full sequence p n 2 n 0 is studied and a careful spectral analysis of the jacobi expression is carried out

canonical forms of self adjoint boundary conditions for differential operators of order four abstract canonical forms of regular self adjoint boundary conditions for differential operators are well known in the second order i e sturm liouville case in this paper we find canonical forms for fourth order self adjoint boundary conditions

regular approximation of linear hamiltonian operators with two singular endpoints abstract in this paper we investigate regular approximation of linear hamiltonian operators with two singular endpoints for any given singular self adjoint hamiltonian operator its spectrum can be approximated with the eigenvalues of a sequence of regular operators the inherited restriction operators are constructed by the limit circle solutions and the limit point solutions and proved to be self adjoint via the abstract operator theory we obtain the strongly resolvent convergence of the inherited regular operators and spectral inclusion of spectrum moreover for the case with both endpoints being limit circle spectral exactness of spectrum is obtained by hilbert schmidt norm convergence in the case of friedrichs extension of singular hamiltonian systems spectral exactness also holds due to variational principle of eigenvalues

singular left definite sturm liouville problems abstract we study singular left definite sturm liouville problems with an indefinite weight function the existence of eigenvalues is established based on the existence of eigenvalues of corresponding right definite problems furthermore for each singular left definite problem with limit circle non oscillatory endpoints we construct a regular left definite problem with the same eigenvalues and use it to obtain properties of eigenvalues and eigenfunctions inequalities among eigenvalues recently established for regular left definite problems are extended to the singular case

necessary conditions for schur stability of interval polynomials new bounds on the coefficient diameter of real schur stable interval polynomials are given using techniques from complex analysis they can be used to unmask interval polynomials at low computational cost as being non schur stable

a general left definite theory for certain self adjoint operators with applications to differential equations we show that any self adjoint operator a bounded or unbounded in a hilbert space h v that is bounded below generates a continuum of hilbert spaces hr r 0 and a continuum of self adjoint operators ar r 0 for reasons originating in the theory of differential operators we call each hr the rth left definite space and each ar the rth left definite operator associated with h a each space hr can be seen as the closure of the domain d ar of the self adjoint operator ar in the topology generated from the inner product arx y x y d ar furthermore each ar is a unique self adjoint restriction of a in hr we show that the spectrum of each ar agrees with the spectrum of a and the domain of each ar is characterized in terms of another left definite space the hilbert space spectral theorem plays a fundamental role in these constructions we apply these results to two examples including the classical laguerre differential expression l in which we explicitly find the left definite spaces and left definite operators associated with a the self adjoint operator generated by l in l2 0 t e t having the laguerre polynomials as eigenfunctions

properties and decompositions of domains for powers of the jacobi differential operator abstract we set out to build a framework for self adjoint extension theory for powers of the jacobi differential operator that does not make use of classical deficiency elements instead we rely on simpler functions that capture the impact of these elements on extensions but are defined by boundary asymptotics this new perspective makes calculations much more accessible and allows for a more nuanced analysis of the associated domains the maximal domain for n th composition of the jacobi operator is characterized in terms of a smoothness condition for each derivative and the endpoint behavior of functions in the underlying hilbert space can then be classified for j n 0 by 1 x j 1 x j 1 x j and 1 x j most of these behaviors can only occur when functions are in the associated minimal domain and this leads to a formulation of the defect spaces with a convenient basis self adjoint extensions including the important left definite domain are then given in terms of the new basis functions for the defect spaces using gkn theory comments are made for the laguerre operator as well

a spectral transform method for singular sturm liouville problems with applications to energy diffusion in plasma physics we develop a spectrally accurate numerical method to compute solutions of a model pde used in plasma physics to describe diffusion in velocity space due to fokker planck collisions the solution is represented as a discrete and continuous superposition of normalizable and nonnormalizable eigenfunctions via the spectral transform associated with a singular sturm liouville operator we present a new algorithm for computing the spectral density function of the operator that uses chebyshev polynomials to extrapolate the value of the titchmarsh weyl m function from the complex upper half plane to the real axis the eigenfunctions and density function are rescaled and a new formula for the limiting value of the m function is derived to avoid amplification of roundoff errors when the solution is reconstructed the complexity of the algorithm is also analyzed showing that the cost of computing the spectral density function at a point grows less rapidly than any fractional inverse power of the desired acc

the left definite spectral theory for the dunkl hermite differential difference equation abstract in this paper we develop the left definite spectral theory associated with the self adjoint operator a in l 2 r t 2 exp t 2 generated from the dunkl second order hermite differential equation l y t t 2 y t 2tt y t 2 y t y t ky t y t r that has the generalized hermite polynomials hm m 0 as eigenfunctions and where t is a differential difference operator called the dunkl operator on r of index more specifically for each n n we explicitly determine the unique left definite hilbert space wn and associated inner product n which is generated from the nth integral power l n of l moreover for each n n we determine the corresponding unique left definite self adjoint operator a n in wn and characterize its domain in terms of another left definite space as a consequence we explicitly determine the domain of each integral power of a and in particular we obtain a new characterization of the domain of the dunkl right definite operator a

symmetrical differential operators and their friedrichs extension abstract symmetric operator realizations of ordinary regular differential expressions are characterized explicitly by boundary conditions for any such operator which is bounded below the boundary condition determining its friedrichs extension is identified

semi boundedness of ordinary differential operators abstract the boundedness from above and from below of very general symmetric quasi differential operators s is studied here we show that i if s is regular of even order and has positive leading coefficient then it is bounded below ii if the order of s which may be regular or singular is odd then s is unbounded both above and below and iii if s is of even order regular or singular but with a leading coefficient which changes sign then s is unbounded both above and below

new results on the bochner condition about classical orthogonal polynomials the classical polynomials hermite laguerre bessel and jacobi are the only orthogonal polynomial sequences ops whose elements are eigenfunctions of the bochner second order differential operator f bochner 1929 3 in loureiro maroni and da rocha 2006 18 these polynomials were described as eigenfunctions of an even order differential operator fk with polynomial coefficients defined by a recursive relation here an explicit expression of fk for any positive integer k is given the main aim of this work is to explicitly establish sums relating any power of f with fk k 1 in other words to bring a pair of inverse relations between these two operators this goal is accomplished with the introduction of a new sequence of numbers the so called a modified stirling numbers which could be also called as bessel or jacobi stirling numbers depending on the context and the values of the complex parameter a

dependence of eigenvalues of a class of fourth order sturm liouville problems on the boundary in this paper we consider the dependence of eigenvalues of a class of fourth order sturm liouville problems on the boundary we show that the eigenvalues depend not only continuously but smoothly on boundary points and that the derivative of the nth eigenvalue as a function of an endpoint satisfies a first order differential equation in addition we prove that as the length of the interval shrinks to zero all higher fourth order dirichlet eigenvalues march off to plus infinity this is also true for the first i e lowest eigenvalue

the classification of self adjoint boundary conditions of differential operators with two singular endpoints abstract for general even order linear ordinary differential equations with real coefficients and endpoints which are regular or singular and for arbitrary deficiency index d the self adjoint domains are determined by d linearly independent boundary conditions these conditions are of three types separated coupled and mixed we give a construction for all conditions of each type and determine the number of conditions of each type possible for a given self adjoint domain our construction gives a direct alternative to the recent construction of everitt and markus which uses the theory of symplectic spaces we believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self adjoint boundary conditions such forms are known only in the second order i e sturm liouville case even for regular problems of order four no such forms are available in the case when all d conditions are separated this construction yields explicit non real conditions for all orders greater than two it is well known that no such conditions exist in the second order case

singularity of the n th eigenvalue of high dimensional sturm liouville problems abstract it is natural to consider continuous dependence of the n th eigenvalue on d dimensional d 2 sturm liouville problems after the results on 1 dimensional case by kong wu and zettl 14 in this paper we find all the boundary conditions such that the n th eigenvalue is not continuous and give complete characterization of asymptotic behavior of the n th eigenvalue this renders a precise description of the jump phenomena of the n th eigenvalue near such a boundary condition furthermore we divide the space of boundary conditions into 2 d 1 layers and show that the n th eigenvalue is continuously dependent on sturm liouville equations and on boundary conditions when restricted into each layer in addition we prove that the analytic and geometric multiplicities of an eigenvalue are equal finally we obtain derivative formula and positive direction of eigenvalues with respect to boundary conditions

a survey of extreme point results for robustness of control systems abstract this paper surveys a subset of the body of research which was sparked by kharitonov s theorem the focal point is extreme point results for robust stability and robust performance that is we give conditions under which satisfaction of a performance specification is ascertained for a family of systems by only checking a finite subset of the extreme members of this family the results which are surveyed apply mainly to systems with structured real parametric uncertainty in addition a number of counterexamples are given to illustrate cases for which an extreme point result does not hold for such cases a solution via the so called edge theorem is often possible

matrices with prescribed lower triangular part abstract we give a necessary and sufficient condition for the existence of a square matrix with prescribed characteristic polynomial and prescribed entries on and below the main diagonal

jacobi stirling numbers jacobi polynomials and the left definite analysis of the classical jacobi differential expression we develop the left definite analysis associated with the self adjoint jacobi operator generated from the classical second order jacobi differential expression in the hilbert space where w t 1 t 1 t that has the jacobi polynomials as eigenfunctions here 1 and k is a fixed non negative constant more specifically for each we explicitly determine the unique left definite hilbert sobolev space and the corresponding unique left definite self adjoint operator in associated with the pair the jacobi polynomials form a complete orthogonal set in each left definite space and are the eigenfunctions of each moreover in this paper we explicitly determine the domain of each as well as each integral power of the key to determining these spaces and operators is in finding the explicit lagrangian symmetric form of the integral composite powers of l k in turn the key to determining these powers is a double sequence of numbers which we introduce in this paper as the jacobi stirling numbers some properties of these numbers which in some ways behave like the classical stirling numbers of the second kind are established including a remarkable and yet somewhat mysterious identity involving these numbers and the eigenvalues of

the laguerre type operator in a left definite hilbert space abstract this article completes earlier work concerning the laguerre type differential operator of fourth order set in a weighted sobolev space on 0 it is shown that the operator which is self adjoint in l 2 0 e x r is also self adjoint in the new space whose inner product also involves first and second derivatives further the spectrum remains unchanged the laguerre type polynomials are eigenfunctions and the spectral resolution associated with the operator is still an eigenfunction expansion

eigenvalues of a class of regular fourth order sturm liouville problems abstract the eigenvalues of a regular fourth order sturm liouville sl problems depend not only continuously but smoothly on the problem an expression for the derivative of the eigenvalues with respect to a given parameter an endpoint a boundary condition a coefficient or the weight function are found

matrix representations of fourth order boundary value problems with finite spectrum abstract we show that a class of regular self adjoint fourth order boundary value problems is equivalent to a certain class of matrix problems equivalent here means that they have exactly the same eigenvalues such an equivalence was previously known only in the second order case

some discrete time counterparts to kharitonov s stability criterion for uncertain systems in 1 kharitonov gave an elegant and simple stability criterion for continuous time systems this note reports on similar results for discrete time systems

left definite theory with applications to orthogonal polynomials in the past several years there has been considerable progress made on a general left definite theory associated with a self adjoint operator a that is bounded below in a hilbert space h the term left definite has its origins in differential equations but littlejohn and wellman l l littlejohn r wellman a general left definite theory for certain self adjoint operators with applications to differential equations j differential equations 181 2 2002 280 339 generalized the main ideas to a general abstract setting in particular it is known that such an operator a generates a continuum h r r 0 of hilbert spaces and a continuum of a r r 0 of self adjoint operators in this paper we review the main theoretical results in l l littlejohn r wellman a general left definite theory for certain self adjoint operators with applications to differential equations j differential equations 181 2 2002 280 339 moreover we apply these results to several specific examples including the classical orthogonal polynomials of laguerre hermite and jacobi

real parameter square integrable solutions and the spectrum of differential operators abstract we continue to investigate the connection between the spectrum of self adjoint ordinary differential operators with arbitrary deficiency index d and the number of linearly independent square integrable solutions for real values of the spectral parameter we show that if for all in an open interval i there are d linearly independent square integrable solutions then there is no continuous spectrum in i this for any self adjoint realization with boundary conditions which may be separated coupled or mixed the proof is based on a new characterization of self adjoint domains and on limit point lp and limit circle lc solutions established in an earlier paper

on j selfadjoint extensions of j symmetric ordinary differential operators on donne une caracterisation des conditions aux limites qui determinent le domaine d une extension j autoadjointe d un operateur differentiel j symetrique t 0 qui a une extremite singuliere

on interval polynomials with no zeros in the unit disc we give a necessary condition for an interval polynomial to have no zeros in the closed unit disc the condition is expressed in terms of the two first intervals

left definite theory for second order differential operators with mixed boundary conditions abstract left definite theory of regular self adjoint operators in a sobolev space was developed a few years ago when the boundary conditions were separated the separation being necessary in order to properly define the sobolev inner product we show how this may be extended when the boundary conditions are not separated but when evaluations at both ends are mixed together there are essentially three cases which arise first when a coefficient matrix is nonsingular second when it is singular but not 0 third when it is 0 the middle case does not arise under separated conditions

accurate spectral numerical schemes for kinetic equations with energy diffusion we examine the merits of using a family of polynomials that are orthogonal with respect to a non classical weight function to discretize the speed variable in continuum kinetic calculations we consider a model one dimensional partial differential equation describing energy diffusion in velocity space due to fokker planck collisions this relatively simple case allows us to compare the results of the projected dynamics with an expensive but highly accurate spectral transform approach it also allows us to integrate in time exactly and to focus entirely on the effectiveness of the discretization of the speed variable we show that for a fixed number of modes or grid points the non classical polynomials can be many orders of magnitude more accurate than classical hermite polynomials or finite difference solvers for kinetic equations in plasma physics we provide a detailed analysis of the difference in behavior and accuracy of the two families of polynomials for the non classical polynomials if the initial condition is not smooth at the origin when interpreted as a three dimensional radial function the exact solution leaves the polynomial subspace for a time but returns up to roundoff accuracy to the same point evolved to by the projected dynamics in that time by contrast using classical polynomials the exact solution differs significantly from the projected dynamics solution when it returns to the subspace we also explore the connection between eigenfunctions of the projected evolution operator and non normalizable eigenfunctions of the full evolution operator as well as the effect of truncating the computational domain

complex j symplectic geometry characterization for j symmetric extensions of j symmetric differential operators we give complex j symplectic geometry characterizations for j symmetric extensions of j symmetric ordinary differential operators

maximal operators of exotic and non exotic laguerre and other semigroups associated with classical orthogonal expansions classical settings of discrete and continuous orthogonal expansions like those of laguerre bessel and jacobi are associated with second order differential operators playing the role of the laplacian these depend on certain type parameters that are usually restricted to a half line or a product of half lines if higher dimensions are considered following earlier research done by hajmirzaahmad we deal in this paper with laplacians in the above mentioned contexts with no restrictions on the type parameters and bring to attention naturally associated orthogonal systems that in fact involve the classical ones but are different this reveals new frameworks related to classical orthogonal expansions and thus new potentially rich research areas at least from the harmonic analysis perspective to support the last claim we focus on maximal operators of multi dimensional laguerre bessel and jacobi semigroups with unrestricted type parameters and prove that they satisfy weak type 1 1 estimates with respect to the appropriate measures generally these measures are not locally finite which makes a contrast with the classical situations and generates new difficulties a significant result of the paper is a new proof of the weak type 1 1 estimate for the classical multi dimensional laguerre semigroup maximal operator

hilbert scales and sobolev spaces defined by associated legendre functions in this paper we study the hilbert scales defined by the associated legendre functions for arbitrary integer values of the parameter this problem is equivalent to studying the left definite spectral theory associated to the modified legendre equation we give several characterizations of the spaces as weighted sobolev spaces and prove identities among the spaces corresponding to the lower regularity index

the higher order differential operators in direct sum spaces w n everitt and a zettl l introduced the concept of differential operator into the direct sum space and provided an operator theoretic framework for the study of two generalized sturm liouville differential operators together m defined on an interval i and m defined on i in particular they characterized self adjoint extensions of the minimal operator in the direct sum space in terms of boundary conditions the aim of this paper is to study the same problem for the higher order differential operators in the direct sum spaces since sturm liouville differential operators always have equal deficiency indices the minimal operators in the direct sum spaces also have equal deficiency indices and self adjoint extensions but for higher order differential operators on an interval z as is well known the values of the deficiency index are very complicated e g see 7 chap 61 hence the minimal operator in the direct sum space has equal deficiency indices only when the deficiency indices of two higher order differential operators satisfy certain conditions in this paper we use cao s and sun s methods 2 31 to give conditions which characterize the domains of self adjoint extensions of minimal operators