linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

on lagrange interpolations and kramer s sampling theorem associated with self adjoint boundary value problems abstract it is known that kramer s sampling theorem and lagrange type interpolation generalize the celebrated whittaker shannon kotel nikov sampling theorem in two different directions however no direct connection between these two directions seems to be known in this article we show that kramer s sampling theorem gives nothing more than the lagrange type interpolations provided that the kernel function associated with kramer s theorem arises from a self adjoint boundary value problem with sample eigenvalues

on kramer s sampling theorem associated with general strum liouville problems and lagrange interpolation kramer s sampling theorem which is a generalization of shannon s sampling theorem and its relationship with lagrange interpolation is studied kramer s sampling theorem is extended to the case where the interval a b is infinite and it is shown that any function that has a sampling expansion in the scope of kramer s theorem also has a lagrange type interpolation expansion provided that the kernel associated with kramer s theorem arises from a second order sturm liouville boundary value problem whether it is regular or singular this new technique not only reproduces many of the known sampling expansions such as those for the cosine the finite hankel and the continuous legendre transforms but also generates new ones such as those for the continuous jacobi laguerre and hermite transforms

basic sturm liouville problems this paper is devoted to studying a q analogue of sturm liouville eigenvalue problems we formulate a self adjoint q difference operator in a hilbert space some of the properties of the eigenvalues and the eigenfunctions are discussed green s function is constructed and the problem in question is inverted into a q type fredholm integral operator with a symmetric kernel the set of eigenfunctions is shown to be a complete orthogonal set in the hilbert space examples involving basic trigonometric functions are involved

on sampling theory and basic sturm liouville systems we investigate the sampling theory associated with basic sturm liouville eigenvalue problems we derive two sampling theorems for integral transforms whose kernels are basic functions and the integral is of jackson s type the kernel in the first theorem is a solution of a basic difference equation and in the second one it is expressed in terms of basic green s function of the basic sturm liouville systems examples involving basic sine and cosine transforms are given

a problem of l l campbell on the equivalence of the kramer and shannon sampling theorems abstract a result concerning the equivalence of the sampling theorems of kramer and whittaker shannon kotelnikov is presented in case the associated kernels are bessel functions or jacobi functions the general problem is open a generalized mehler dirichlet formula for the jacobi functions is used to give the proof in this case for the bessel functions a generalization of poisson s integral is needed it turns out that in both cases kramer s theorem gives nothing more than shannon s theorem in the sense that each function that can be sampled by kramer s theorem can also be reconstructed by the classical shannon sampling result

on the inversion of integral transforms associated with sturm liouville problems consider the sturm liouville boundary value problem 1 1 y q x y t2y a x b 2 2 y a cos y a sin 0 3 3 y b cos y b sin 0 where q x is continuous on a b let x t be a solution of either the initial value problem 1 and 2 or 1 and 3 in this paper we develop two techniques to invert the integral f t abf x x t dx where f x l2 a b one technique is based on the construction of some biorthogonal sequence of functions and the other is based on poisson s summation formula

a computation of fermi dirac integrals by asymptotics and the hermite corrector formula we present a computation of fermi dirac integrals based on asymptotic expansions and the hermite corrector formula

sampling of vector valued transforms associated with green s matrix of dirac systems recently a sampling theorem associated with regular one dimensional dirac systems has been developed integral transforms whose kernels are solutions of the systems have been reconstructed from their values at the eigenvalues in the present article the situation is considered where the kernels are replaced by green s matrix of the problem thus a vector valued sampling theorem is established for vector valued transforms a family of examples where the systems have variable coefficients and a vector valued kramer type sampling lemma are given at the end of the paper

on lagrange interpolation and kramer type sampling theorems associated with sturm liouville problems this article is devoted to a connection between kramer s sampling theorem and sampling expansions generated by lagrange interpolation it is shown that any function that has a sampling expansion in the scope of kramer s theorem also has a lagrange type interpolation expansion provided that the kernel associated with kramer s theorem arises from a second order sturm liouville boundary value problem this new approach which for a variety of regular and singular sturm liouville problems leads to associated sampling theorems recovers not only many known sampling expansions but also gives new ways to calculate the corresponding sampling functions new sampling series are included

bounded rank one perturbations in sampling theory sampling theory concerns the problem of reconstruction of functions from the knowledge of their values at some discrete set of points in this paper we derive an orthogonal sampling theory and associated lagrange interpolation formulae from a family of bounded rank one perturbations of a self adjoint operator that has only discrete spectrum of multiplicity one

sampling expansions associated with kamke problems the present paper is devoted to the derivation of sampling expansions for entire functions which are represented as integral transforms where a differential operator is acting on the kernels the situation generalizes the results obtained in sampling theory associated with boundary value problems to the case when the differential equation has the form n y lambda p y where n and p are two differential expressions of orders n and p respectively n p and lambda is the eigenvalue parameter both self adjoint and non self adjoint cases will be considered with examples in which the boundary conditions are strongly regular

maps on matrix spaces it is well known that every automorphism of the full matrix algebra is inner we give a short proof of this statement and discuss several extensions of this theorem including structural results for multiplicative maps on matrix algebras characterizations of monotone and orthogonality preserving maps on idempotent matrices some nonlinear preserver results and some recent theorems concerning geometry of matrices we show that all these topics are closely related and point out the connections with physics and geometry several open problems are posed

a numerical evaluation of very oscillatory infinite integrals by asymptotics recently levin and sidi have given some nonlinear methods for the accurate evaluation of slowly converging infinite integrals but one needs to solve some very instable linear systems when for example the integrand have an increasingly rapid oscillatory behavior at infinity in this work we present a direct method which avoids the instable linear systems

on hua s fundamental theorem of the geometry of rectangular matrices abstract we show that hua s fundamental theorem of the geometry of rectangular matrices can be proved without the bijectivity assumption when the underlying field is the field of real numbers we also give a counterexample showing that this generalization is not possible in the complex case

spectrum and commutativity preserving mappings on triangular matrices we give the description of continuous mappings on the algebra of all upper triangular complex matrices that preserve spectrum and commutativity

sampling theory associated with q difference equations of the sturm liouville type we will show that if the kernel k x t in the representation f t a0k x t u t dqt with u l2q 0 a is a solution of a second order q sturm liouville boundary problem then f admits a representation as a sampling formula of the form f t n 0f n w t w n t n where n is the nth eigenvalue of the associated q sturm liouville boundary problem and w t is the q wronskian of two solutions selected in a specified way

applications of krein s theory of regular symmetric operators to sampling theory the classical kramer sampling theorem establishes general conditions that allow the reconstruction of functions by mean of orthogonal sampling formulae one major task in sampling theory is to find concrete non trivial realizations of this theorem in this paper we provide a new approach to this subject on the basis of krein s theory of representation of simple regular symmetric operators having deficiency indices 1 1 we show that the resulting sampling formulae have the form of lagrange interpolation series we also characterize the space of function reconstructible by our sampling formulae our construction allows a rigorous treatment of certain ideas proposed recently in quantum gravity

mappings preserving spectrum and commutativity on hermitian matrices the general form of a continuous mapping acting on the real vector space of all n n complex hermitian or real symmetric matrices and preserving spectrum and commutativity is derived it turns out that is either linear or its image forms a commutative set

sampling theorem associated with a dirac operator and the hartley transform abstract kramer s sampling theorem indicates that under certain conditions sampling theorems associated with boundary value problems involving n th order self adjoint differential operators can be obtained in this paper we extend this result and derive a sampling theorem associated with boundary value problems involving a one dimensional system of dirac operators as a special case we obtain a sampling theorem for the hartley transform of a bandlimited function

hua s fundamental theorems of the geometry of matrices and related results abstract we briefly survey some recent improvements of hua s fundamental theorem of the geometry of rectangular matrices then we discuss possible further generalizations as well as some related open problems in the theory of preservers we solve one such open problem using ovchinnikov s characterization of automorphisms of the poset of idempotent matrices using ovchinnikov s result we obtain a short proof of the fundamental theorem of the geometry of square matrices

oversampling and aliasing in de branges spaces arising from bessel operators abstract we show that a class of de branges spaces generated by means of generalized fourier transforms associated with perturbed bessel differential equations has the properties of oversampling and aliasing

sampling in paley wiener spaces associated with fractional integro differential operators this paper employs the theory of kramer analytic kernels established by everitt et al results in math 34 1998 310 to derive sampling expansions the kernels of the recovered transforms are mittag leffler functions of two parameters these kernels are solutions of certain classes of fractional integro differential equations the technique used here is different from the analytic approach by djrbashian harmonic analysis and boundary value problems in the complex domain birkhauser basel 1993

characterization of jordan homomorphisms on mn using preserving properties abstract let o mn mn n 3 be a continuous mapping preserving spectrum and commutativity in both directions neither linearity nor multiplicativity of i is assumed then i is either an automorphism or an antiautomorphism the same result holds if i preserves spectrum commutativity and rank one matrices

sampling theorems associated with fourth and higher order self adjoint eigenvalue problems abstract consider a regular self adjoint eigenvalue problem of n th order and the associated whittaker shannon kramer sampling series the samples being taken at the eigenvalues and the sinc function being replaced by the generalized fourier coefficients of the kernel arising from the eigenvalue problem with respect to the complete set of orthogonal eigenfunctions the major aim of this paper is to show that each such sampling expansion can be written as a lagrange interpolation series provided the kernel satisfies suitable conditions two concrete fourth order eigenvalue problems are examined a regular one as an application of the general result and a singular one the problem whether the two resulting sampling series are really new or can be reduced to already known results is in part open

spectrum and commutativity preserving mappings on h2 abstract the continuous mappings on 2 2 real symmetric or complex hermitian matrices that are spectrum and commutativity preserving are characterized