matricial nehari problems j inner matrix functions and the muckenhoupt condition abstract the classes of regular and strongly regular generating matrices and j inner matrix valued functions arose in the investigation of the matricial nehari problem bitangential interpolation problems and inverse problems for canonical systems as well as the theory of characteristic functions of operators and operator nodes in this paper new characterizations of these classes are developed in particular the property of strong regularity is characterized in terms of a matricial muckenhoupt a2 condition in the treil volberg form these results are based on parametrizations that are intimately connected with darlington representations of matrix valued functions in the schur and caratheodory classes as a byproduct of this analysis examples of strongly regular generating matrices and entire j inner matrix valued functions that are unbounded on the circle and the real line respectively are presented

on the zeros of some continuous analogues of matrix orthogonal polynomials and a related extension problem with negative squares a new proof of a recent theorem of ellis gohberg and lay which identifies the number of roots of a continuous matrix orthogonal polynomial in the open upper halfplane with the number of negative eigenvalues of a related integral operator is presented a related extension problem is then formulated and solved in assorted classes of functions which are analytic in the open upper half plane apart from a finite number of poles a discrete analogue of this extension problem is also formulated and solved 1994 john wiley sons inc

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

on the inverse scattering problem for jacobi matrices with the spectrum on an interval a finite system of intervals or a cantor set of positive length when solving the inverse scattering problem for a discrete sturm liouville operator with a rapidly decreasing potential one gets reflection coefficients s and invertible operators where is the hankel operator related to the symbol s the marchenko faddeev theorem 8 in the continuous case for the discrete case see 4 6 guarantees the uniqueness of the solution of the inverse scattering problem in this article we ask the following natural question can one find a precise condition guaranteeing that the inverse scattering problem is uniquely solvable and that operators are invertible can one claim that uniqueness implies invertibility or vise versa moreover we are interested here not only in the case of decreasing potential but also in the case of asymptotically almost periodic potentials so we merge here two mostly developed cases of the inverse problem for sturm liouville operators the inverse problem with almost periodic potential and the inverse problem with the fast decreasing potential

structured invariant spaces of vector valued rational functions hermitian matrices and a generalization of the lohvidov laws abstract finite dimensional indefinite inner product spaces of vector valued rational functions which are 1 invariant under the generalized backward shift and 2 subject to a structural identity and subspaces and superspaces thereof are studied the theory of these spaces is then applied to deduce a generalization of a pair of rules due to lohvidov for evaluating the inertia of certain subblocks of hermitian toeplitz and hermitian hankel matrices the connecting link rests on the identification of a hermitian matrix as the gram matrix of a space of vector valued functions of the type considered in the first part of the paper corresponding generalizations of another pair of theorems by lohvidov on the rank of certain subblocks of non hermitian teoplitz and non hermitian hankel matrices are also stated but the proofs will be presented elsewhere

on a new class of reproducing kernel spaces and a new generalization of the iohvidov laws abstract a new class of finite dimensional reproducing kernel spaces of m 1 vector valued analytic functions on a fairly general domain is introduced the reproducing kernels of these spaces have a special form which is based on an m m matrix valued function which is j unitary on the boundary of every invertible hermitian matrix can be interpreted as the gram matrix of a suitably chosen basis in such a space where m is equal to the appropriately defined displacement rank of the given matrix orthogonal decompositions of the space are developed in terms of schur complements of the matrix and factorizations of the much as in the classical cases wherein is either the unit disc or the half plane finally a new generalization of the iohvidov laws which extends earlier generalizations by the same authors is deduced as an application of the theory

caratheodory fejer interpolation in the ball abstract we use the theory of reproducing kernel hilbert spaces to solve a caratheodory fejer interpolation problem in the class of schur multipliers of the reproducing kernel hilbert space of functions analytic in the unit ball of c n with reproducing kernel 1 1 1 n z k w k

universal algorithm for optimal estimation of quantum states from finite ensembles via realizable generalized measurement we present a universal algorithm for the optimal quantum state estimation of an arbitrary finite dimensional system the algorithm specifies a physically realizable i e finite positive operator valued measurement on a finite number of identically prepared systems we illustrate the general formalism by applying it to different scenarios of the state estimation of n independent and identically prepared two level systems qubits

rational matrix functions with coisometric values on the imaginary line abstract given signature matrices j1 and j2 we obtain a necessary and sufficient condition for a rational matrix function w analytic at infinity to satisfy equation j1 w z j2w z on the imaginary axis the condition is based on a lyapunov equation involving matrices in an observable realization of w and generalizes the fact well known in the case where j1 j2 if the condition is satisfied for every observable realization a b c d of w there exists a unique possibly singular hermitian matrix g such that g satisfies the lyapunov equation and cg dj2b we call g the hermitian matrix associated with the realization the minimal factorizations w w1w2 where w1 and w2 satisfy equations j1 w1 z j1w1 z and j1 w2 z j2w2 z for z on the imaginary line can be characterized in terms of decompositions of the state space into subspaces determined by a possibly indefinite inner product induced by g as a corollary we obtain a sufficient condition for existence of a minimal factorization w w1w2 with w1 the multiplicative inverse of a blaschke potapov factor

on a new class of structured reproducing kernel spaces abstract a class of reproducing kernel spaces with reproducing kernels of the form k j j with p a a is characterized in terms of invariance under a pair of generalized shift operators and a structural identity this incorporates a characterization of de branges for the line case and a later analogue due to ball for the circle case as well as many other possibilities by specializing the choice of these results also permit the extension of some earlier characterizations by the authors of finite dimensional spaces with reproducing kernels of the form given above to the infinite dimensional case the non hermitian case is also considered

on a general moment problem and certain matrix equations abstract let k be a nonnegative matrix satisfying a matrix equation of the form a k a b k b m jm where a b and m are of a special form we look for representations of k of the form k 0 x di x where 0 is a simple closed contour di is a positive matrix valued measure and the matrix valued function x depending on a b and m defines a space of functions invariant with respect to a shift associated to 0 the method used is that of the fundamental matrix inequality of potapov suitably adapted to the present framework

the schur algorithm and reproducing kernel hilbert spaces in the ball abstract using reproducing kernel hilbert spaces methods we develop a schur type algorithm for a subclass of the functions analytic and contractive in the ball we also consider the nevanlinna pick interpolation problem in that class

universal multi task kernels in this paper we are concerned with reproducing kernel hilbert spaces hk of functions from an input space into a hilbert space y an environment appropriate for multi task learning the reproducing kernel k associated to hk has its values as operators on y our primary goal here is to derive conditions which ensure that the kernel k is universal this means that on every compact subset of the input space every continuous function with values in y can be uniformly approximated by sections of the kernel we provide various characterizations of universal kernels and highlight them with several concrete examples of some practical importance our analysis uses basic principles of functional analysis and especially the useful notion of vector measures which we describe in sufficient detail to clarify our results

the symbol grounding problem there has been much discussion recently about the scope and limits of purely symbolic models of the mind and about the proper role of connectionism in cognitive modeling this paper describes the symbol grounding problem how can the semantic interpretation of a formal symbol system be made intrinsic to the system rather than just parasitic on the meanings in our heads how can the meanings of the meaningless symbol tokens manipulated solely on the basis of their arbitrary shapes be grounded in anything but other meaningless symbols the problem is analogous to trying to learn chinese from a chinese chinese dictionary alone a candidate solution is sketched symbolic representations must be grounded bottom up in nonsymbolic representations of two kinds 1 iconic representations which are analogs of the proximal sensory projections of distal objects and events and 2 categorical representations which are learned and innate feature detectors that pick out the invariant features of object and event categories from their sensory projections elementary symbols are the names of these object and event categories assigned on the basis of their nonsymbolic categorical representations higher order 3 symbolic representations grounded in these elementary symbols consist of symbol strings describing category membership relations e g an x is a y that is z connectionism is one natural candidate for the mechanism that learns the invariant features underlying categorical representations thereby connecting names to the proximal projections of the distal objects they stand for in this way connectionism can be seen as a complementary component in a hybrid nonsymbolic symbolic model of the mind rather than a rival to purely symbolic modeling such a hybrid model would not have an autonomous symbolic module however the symbolic functions would emerge as an intrinsically dedicated symbol system as a consequence of the bottom up grounding of categories names in their sensory representations symbol manipulation would be governed not just by the arbitrary shapes of the symbol tokens but by the nonarbitrary shapes of the icons and category invariants in which they are grounded

operators affiliated to the free shift on the free hardy space abstract the smirnov class for the classical hardy space is the set of ratios of bounded analytic functions in the open complex unit disk with outer denominators this definition extends naturally to the commutative and non commutative multi variable settings of the drury arveson space and the full fock space over c d identifying the fock space with the free multi variable hardy space of non commutative or free holomorphic functions in a non commutative open unit ball in several matrix variables we prove that any closed densely defined operator affiliated to the right free multiplier algebra of the full fock space acts as right multiplication by a non commutative function in the right free smirnov class and analogously replacing right with left

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

gleason s problem and homogeneous interpolation in hardy and dirichlet type spaces of the ball we solve gleason s problem in the reproducing kernel hilbert spaces with reproducing kernels 1 1 1nzjwj r for real r 0 and their counterparts for r 0 and study the homogeneous interpolation problem in these spaces

backward shift operator and finite dimensional de branges rovnyak spaces in the ball recently bolotnikov and rodman characterized finite dimensional backward shift invariant subspaces of the arveson space in this note we study finite dimensional backward shift invariant hilbert spaces contractively included in the arveson space to that purpose we redefine in an appropriate way the backward shift operators in terms of power series expansions

criteria for the strong regularity of j inner functions and γ generating matrices abstract the class of left and right strongly regular j inner mvf s plays an important role in bitangential interpolation problems and in bitangential direct and inverse problems for canonical systems of integral and differential equations a new criterion for membership in this class is presented in terms of the matricial muckenhoupt condition a 2 that was introduced for other purposes by treil and volberg analogous results are also obtained for the class of generating functions that intervene in the nehari problem the new criterion is simpler than the criterion that we presented earlier a determinental criterion is also presented

boundary interpolation in the ball we solve a boundary interpolation problem in the reproducing kernel hilbert space of functions analytic in the unit ball of cn with reproducing kernel 1 1 1nzkwk we introduce the notion of brune factor or blaschke potapov factor of the third kind in this setting

geometric learning algorithms abstract emergent computation in the form of geometric learning is central to the development of motor and perceptual systems in biological organisms and promises to have a similar impact on emerging technologies including robotics vision speech and graphics this paper examines some of the trade offs involved in different implementation strategies focusing on the tasks of learning discrete classifications and smooth nonlinear mappings the trade offs between local and global representations are discussed a spectrum of distributed network implementations are examined and an important source of computational inefficiency is identified efficient algorithms based on k d trees and the delaunay triangulation are presented and the relevance to biological networks is discussed finally extensions of both the tasks and the implementations are given

representations of semigroups associated to invariant kernels with values adjointable operators abstract we consider positive semidefinite kernels valued in the algebra of adjointable operators on a ve space vector euclidean space and that are invariant under actions of semigroups a rather general dilation theorem is stated and proved for these kind of kernels representations of the semigroup on either the ve spaces of linearisation of the kernels or on their reproducing kernel ve spaces are obtainable we point out the reproducing kernel fabric of dilation theory and we show that the general theorem unifies many dilation results at the non topological level

on a general moment problem on the half axis abstract we consider the integral representations on the half axis for a pair of nonnegative matrices which satisfy certain matrix equations the problem of describing all such representations generalizes the classical stieltjes moment problem

spectral factorization of rectangular rational matrix functions with application to discrete wiener hopf equations the properties of a discrete wiener hopf equation are closely related to the factorization of the symbol of the equation we give a necessary and sufficient condition for existence of a canonical wiener hopf factorization of a possibly nonregular rational matrix function w relative to a contour which is a positively oriented boundary of a region in the finite complex plane the condition involves decomposition of the state space in a minimal realization of w and if it is satisfied we give explicit formulas for the factors the results are generalized by means of centered realizations to arbitrary rational matrix functions the proposed approach can be used to solve discrete wiener hopf equations whose symbols are rational matrix functions which admit canonical factorization relative to the unit circle

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

reproducing kernel hilbert c modules and kernels associated with cocycles in this paper we discuss reproducing kernels whose ranges are contained in a c algebra or a hilbert c module using the construction of a reproducing hilbert c module associated with a reproducing kernel we show how such a reproducing kernel can naturally be expressed in terms of operators on a hilbert c module using representations on hilbert c modules we prove that for each positive definite g kernel associated with a cocycle there is a representation associated with an operator valued cocycle on the corresponding hilbert c module finally some examples will be considered

classical and quantum probability we follow the development of probability theory from the beginning of the last century emphasizing that quantum theory is really a generalization of this theory the great achievements of probability theory such as the theory of processes generalized random fields estimation theory and information geometry are reviewed their quantum versions are then described

hilbert c modules and projective representations associated with multipliers abstract in this paper we consider projective representations and give the dilations associated with the representations to unify the various representation theorem we use the kolmogorov decompositions for positive definite kernels we present dilation theorems for projective representations which generalize stinespring dilation theorem for covariant completely positive maps as well as naimark sz nagy characterization of positive definite functions we also construct a representation of a right hilbert c module for a strictly u covariant and completely positive map

zero cancellation for general rational matrix functions the problem of cancelling a specified part of the zeros of a completely general rational matrix function by multiplication with an appropriate invertible rational matrix function is investigated from different standpoints firstly the class of all factors that dislocate the zeros and feature minimal mcmillan degree are derived further necessary and sufficient existence conditions together with the construction of solutions are given when the factor fulfills additional assumptions like being j unitary or j inner either with respect to the imaginary axis or to the unit circle the main technical tool are centered realizations that deliver a sufficiently general conceptual support to cope with rational matrix functions which may be polynomial proper or improper rank deficient with arbitrary poles and zeros including at infinity a particular attention is paid to the numerically sound construction of solutions by employing at each stage unitary transformations reliable numerical algorithms for eigenvalue assignment and efficient lyapunov equation solvers

wavelets and the angle between past and future the main result the riesz projectionp or equivalently hilbert transformt is bounded in the weighted spacel2 w wherewis a matrix valued weight if and only ifsupi 1 i iw 1 2 1 i iw 1 1 2 wheresupremumis taken over all intervalsi motivation for this problem comes from stationary processes riesz projection is bounded means the angle between past and future of a stationary process with spectral measurewis nonzero in the scalar case the result is the well known hunt muckenhoupt wheeden theorem the main step in our proof is to show that a vector haar system forms an unconditional basis inl2 w as a byproduct of our approach we get some new results about bases of wavelets in weighted spaces in both scalar and vector valued cases

multiplication operators with deficiency indices p p and sampling formulas in reproducing kernel hilbert spaces of entire vector valued functions abstract a number of recent papers have established connections between reproducing kernel hilbert spaces h of entire functions de branges spaces sampling formulas and a class of symmetric operators with deficiency indices 1 1 in this paper analogous connections between reproducing kernel hilbert spaces of entire vector valued functions de branges spaces of entire vector valued functions sampling formulas and symmetric operators with deficiency indices p p are obtained enroute an analog of l de branges characterization of the reproducing kernel hilbert spaces of entire functions that are now called de branges spaces is obtained for the p 1 vector valued case a special class of these de branges spaces of p 1 vector valued entire functions is identified as a functional model for m g krein s class of entire operators with deficiency indices p p

caratheodory fejer interpolation in the ball with mixed derivatives we use reproducing kernel hilbert space methods to solve a caratheodory fejer type interpolation problem with mixed derivatives corresponding to a so called lower inclusive set in the class of schur multipliers

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

estimation of pure qubits on circles gisin and popescu gisin n and popescu s 1999 phys rev lett 83 432 have shown that more information about their direction can be obtained from a pair of anti parallel spins compared to a pair of parallel spins where the first member of the pair which we call the pointer member can point equally along any direction in the bloch sphere they argued that this was due to the difference in dimensionality spanned by these two alphabets of states here we consider similar alphabets but with the first spin restricted to a fixed small circle of the bloch sphere in this case the dimensionality spanned by the anti parallel versus parallel alphabet is now equal however the anti parallel alphabet is found to still contain more information in general we generalize this to having n parallel spins and m anti parallel spins when the pointer member is restricted to a small circle these alphabets again span spaces of equal dimension yet in general more directional information can be found for sets with smaller n m for any fixed total number of spins we find that the optimal povms for extracting directional information in these cases can always be expressed in terms of the fourier basis our results show that dimensionality alone cannot explain the greater information content in antiparallel combinations of spins compared to parallel combinations in addition we describe an locc protocol which extracts optimal directional information when the pointer member is restricted to a small circle and a pair of parallel spins are supplied

on a new class of realization formulas and their application abstract a new set of realization formulas is derived for a class of matrix valued functions w these include the standard realization formulas for rational w observability controllability and minimality are defined and characterized conditions for w to be j 1 j 2 isometric and or j 1 j 2 coisometric with respect to a pair of signature matrices j 1 and j 2 are given in terms of the realizations minimal factorizations for square w are considered and formulas for the factors are deduced associated reproducing kernel spaces of pairs too are discussed

minimal optimal generalized quantum measurements optimal and finite positive operator valued measurements on a finite number n of identically prepared systems have been presented recently with physical realization in mind we propose here optimal and minimal generalized quantum measurements for two level systems we explicitly construct them up to n 7 and verify that they are minimal up to n 5 we finally propose an expression which gives the size of the minimal optimal measurements for arbitrary n

beurling lax type theorems in the complex and quaternionic setting abstract we give a generalization of the beurling lax theorem both in the complex and quaternionic settings we consider in the first case functions meromorphic in the right complex half plane and functions slice hypermeromorphic in the right quaternionic half space in the second case in both settings we also discuss a unified framework which includes both the disk and the half plane for the complex case and the open unit ball and the half space in the quaternionic setting

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

the structure of inner multipliers on spaces with complete nevanlinna pick kernels let k be the reproducing kernel for a hilbert space h k of analytic functions on bd the open unit ball in cd d 1 k is called a complete np kernel if k0 1 and if 1 1 k z is positive definite on bd bd let d be a separable hilbert space and consider h k d h k d and think of it as a space of d valued h k functions a theorem of mccullough and trent j funct anal 178 2000 226 249 partially extends the beurling lax halmos theorem for the invariant subspaces of the hardy space h2 d they show that if k is a complete np kernel and if d is a separable hilbert space then for any scalar multiplier invariant subspace m of h k d there exists an auxiliary hilbert space e and a multiplication operator h k e h k d such that is a partial isometry and m h k e such multiplication operators are called inner multiplication operators and they satisfy the orthogonal projection onto m in this paper we shall show that for many interesting complete np kernels the analogy with the beurling lax halmos theorem can be strengthened we show that for almost every z bd the nontangential limit z of the multiplier bd b e d associated with an inner multiplication operator is a partial isometry and that rank z is equal to a constant almost everywhere the result applies to certain weighted dirichlet spaces and to the space h2d which is determined by the kernel k z 11 z d z bd in particular our result implies that the curvature invariant of arveson proc natl acad sci usa96 1999 11 096 11 099 of a pure contractive hilbert module of finite rank is an integer this answers a question of w arveson proc natl acad sci usa96 1999 11096 11099

riccati inequalities and reproducing kernel hilbert spaces natural connections between positive semidefinite solutions x of homogeneous algebraic riccati equations and finite dimensional reproducing kernel de branges spaces based on a j inner proper rational square matrix valued functions are known in this paper analogous connections between the positive semidefinite solutions x of nonhomogeneous algebraic riccati equations and finite dimensional reproducing kernel hilbert spaces based on rectangular j j coinner proper rational matrix valued functions are developed and are then applied to obtain factorization formulas for in terms of elementary factors enroute formulas for the factors in a version of a theorem of leech are also obtained

the theory and practice of bayesian image labeling integrating disparate sources of information has been recognized as one of the keys to the success of general purpose vision systems image clues such as shading texture stereo disparities and image flows provide uncertain local and incomplete information about the three dimensional scene spatial a priori knowledge plays the role of filling in missing information and smoothing out noise this thesis proposes a solution to the longstanding open problem of visual integration it reports a framework based on bayesian probability theory for computing an intermediate representation of the scene from disparate sources of information the computation is formulated as a labeling problem local visual observations for each image entity are reported as label likelihoods they are combined consistently and coherently on hierarchically structured label trees with a new computationally simple procedure the pooled label likelihoods are fused with the a priori spatial knowledge encoded as markov random fields mrf s the a posteriori distribution of the labelings are thus derived in a bayesian formalism a new inference method highest confidence first hcf estimation is used to infer a unique labeling from the a posteriori distribution unlike previous inference methods based on the mrf formalism hcf is computationally efficient and predictable while meeting the principles of graceful degradation and least commitment the results of the inference process are consistent with both observable evidence and a priori knowledge the effectiveness of the approach is demonstrated with experiments on two image analysis problems intensity edge detection and surface reconstruction for edge detection likelihood outputs from a set of local edge operators are integrated with a priori knowledge represented as an mrf probability distribution for surface reconstruction intensity information is integrated with sparse depth measurements and a priori knowledge coupled mrf s provide a unified treatment of surface reconstruction and segmentation and an extension of hcf implements a solution method experiments using real image and depth data yield robust results the framework can also be generalized to higher level vision problems as well as to other domains

statistical properties of the method of regularization with periodic gaussian reproducing kernel the method of regularization with the gaussian reproducing kernel is popular in the machine learning literature and successful in many practical applications in this paper we consider the periodic version of the gaussian kernel regularization we show in the white noise model setting that in function spaces of very smooth functions such as the infinite order sobolev space and the space of analytic functions the method under consideration is asymptotically minimax in finite order sobolev spaces the method is rate optimal and the efficiency in terms of constant when compared with the minimax estimator is reasonably high the smoothing parameters in the periodic gaussian regularization can be chosen adaptively without loss of asymptotic efficiency the results derived in this paper give a partial explanation of the success of the gaussian reproducing kernel in practice simulations are carried out to study the finite sample properties of the periodic gaussian regularization

the gaussian free field and hadamard s variational formula we relate the gaussian free field on a planar domain to the variational formula of hadamard which explains the change of the green function under a perturbation of the domain this is accomplished by means of a natural integral operator called the hadamard operator associated with a given flow of growing domains

on the tangential interpolation problem for functions the aim of this paper is to solve a matrix valued version of the nevanlinna pick interpolation problem for hi functions we reduce this prob lem to a nevanlinna pick interpolation problem for schur functions and obtain a linear fractional transformation which describes the set of all solutions

complete nevanlinna pick kernels abstract we give a new treatment of quiggin s and mccullough s characterization of complete nevanlinna pick kernels we show that a kernel has the matrix valued nevanlinna pick property if and only if it has the vector valued nevanlinna pick property we give a representation of all complete nevanlinna pick kernels and show that they are all restrictions of a universal complete nevanlinna pick kernel

a gleason kahane żelazko theorem for the dirichlet space abstract we show that every linear functional on the dirichlet space that is non zero on nowhere vanishing functions is necessarily a multiple of a point evaluation continuity of the functional is not assumed as an application we obtain a characterization of weighted composition operators on the dirichlet space as being exactly those linear maps that send nowhere vanishing functions to nowhere vanishing functions we also investigate possible extensions to weighted dirichlet spaces with superharmonic weights as part of our investigation we are led to determine which of these spaces contain functions that map the unit disk onto the whole complex plane

refinement of operator valued reproducing kernels this paper studies the construction of a refinement kernel for a given operator valued reproducing kernel such that the vector valued reproducing kernel hilbert space of the refinement kernel contains that of the given kernel as a subspace the study is motivated from the need of updating the current operator valued reproducing kernel in multi task learning when underfitting or overfitting occurs numerical simulations confirm that the established refinement kernel method is able to meet this need various characterizations are provided based on feature maps and vector valued integral representations of operator valued reproducing kernels concrete examples of refining translation invariant and finite hilbert schmidt operator valued reproducing kernels are provided other examples include refinement of hessian of scalar valued translation invariant kernels and transformation kernels existence and properties of operator valued reproducing kernels preserved during the refinement process are also investigated

a structure theorem for reproducing kernel pontryagin spaces we illustrate a relationship between reproducing kernel spaces and orthogonal polynomials via a general structure theorem the christofell darboux formula emerges as a limit case

quantum stochastic processes for maps on hilbert c modules we discuss pairs of maps where is a map between c algebras and is a module map between hilbert c modules which are generalization of representations of hilbert c modules a covariant version of stinespring s theorem for such a pair is established and quantum stochastic processes constructed from pairs t t of families of such maps are studied we prove that the quantum stochastic process j jt constructed from a quantum dynamical semigroup t is a j map for the quantum stochastic process j jt constructed from the given quantum dynamical semigroup t and that j is covariant if the quantum dynamical semigroup is covariant

on the reproducing kernel of a pontryagin space of vector valued polynomials abstract we give necessary and sufficient conditions under which the reproducing kernel of a pontryagin space of d 1 vector polynomials is determined by a generalized nevanlinna pair of d d matrix polynomials

singular systems a new approach in the time domain a new approach in the time domain is developed for the study of singular linear systems of the form e dot x ax bu y cx with e singular central to the approach is the fundamental triple alpha e a 1 e alphae a 1 b c with a real number satisfying det alpha e a neq 0 controllability and observability matrices are expressed in terms of the fundamental triple new tests for impulse controllability and impulse observability are also established feedback control problems including pole placement decoupling and disturbance localization are studied by use of a modified proportional and derivative feedback of the state in the form of u f alpha x dot x v

factorizations induced by complete nevanlinna pick factors abstract we prove a factorization theorem for reproducing kernel hilbert spaces whose kernel has a normalized complete nevanlinna pick factor this result relates the functions in the original space to pointwise multipliers determined by the nevanlinna pick kernel and has a number of interesting applications for example for a large class of spaces including dirichlet and drury arveson spaces we construct for every function f in the space a pluriharmonic majorant of f 2 with the property that whenever the majorant is bounded the corresponding function f is a pointwise multiplier

on the influence of the kernel on the consistency of support vector machines in this article we study the generalization abilities of several classifiers of support vector machine svm type using a certain class of kernels that we call universal it is shown that the soft margin algorithms with universal kernels are consistent for a large class of classification problems including some kind of noisy tasks provided that the regularization parameter is chosen well in particular we derive a simple sufficient condition for this parameter in the case of gaussian rbf kernels on the one hand our considerations are based on an investigation of an approximation property the so called universality of the used kernels that ensures that all continuous functions can be approximated by certain kernel expressions this approximation property also gives a new insight into the role of kernels in these and other algorithms on the other hand the results are achieved by a precise study of the underlying optimization problems of the classifiers furthermore we show consistency for the maximal margin classifier as well as for the soft margin svm s in the presence of large margins in this case it turns out that also constant regularization parameters ensure consistency for the soft margin svm s finally we prove that even for simple noise free classification problems svm s with polynomial kernels can behave arbitrarily badly

on a bitangential interpolation problem for contractive valued functions on the unit ball we solve the bitangential interpolation problem with a finite number of interpolation nodes for a multivariable analogue of the schur class consisting of matrix valued analytic functions on the ball the interpolation conditions are formulated via a generalized functional calculus with operator argument thereby generalizing in a compact way the simple first order interpolation conditions considered for this class of functions in earlier work the description of all solutions is given via a redheffer transform whose entries are given explicitly in terms of the interpolation data

rigged de branges pontryagin spaces and their application to extensions and embedding abstract de branges pontryagin spaces b e with negative index of entire p 1 vector valued functions based on an entire p 2 p entire matrix valued function e called the de branges matrix are studied an explicit description of these spaces and an explicit formula for the indefinite inner product are presented a characterization of those spaces b e that are invariant under the generalized backward shift operator that extends known results when 0 is given the theory of rigged de branges pontryagin spaces is developed and then applied to obtain an embedding of de branges matrices with negative squares in generalized j inner matrices and selfadjoint extensions of the multiplication operator in b e a formula for factoring an arbitrary generalized j inner matrix valued function into the product of a singular factor and a perfect factor is found analogous to the known factorization formulas for j inner matrix valued functions

de branges spaces on compact riemann surfaces and a beurling lax type theorem abstract using the notion of commutative operator vessels this work investigates de branges rovnyak spaces whose elements are sections of a line bundle of multiplicative half order differentials on a compact real riemann surface as a special case we obtain a beurling lax type theorem in the setting of the corresponding hardy space on a finite bordered riemann surface

invariant subspaces and nevanlinna pick kernels abstract a theorem of beurling lax halmos represents a subspace m of h2 c d the hardy space of analytic functions with values in the hilbert space e and square summable power series invariant for multiplication by z as h2 f where f is a subspace of e and is an inner function with values in l f e when the hardy space is replaced by the hilbert space h k determined by a nevanlinna pick kernel k such as the dirichlet kernel or the row contraction kernel on the ball in c d the blh theorem survives with f an auxiliary hilbert space and a l f e valued function which is inner in the sense that the operator m of multiplication by is a partial isometry under mild additional hypotheses when e c mz the operator of multiplication by z is cellularly indecomposable and has the codimension one property however if m is invariant for mz m mz m need not be a cyclic subspace for mz restricted to m

monotone quantum stochastic processes and covariant dynamical hemigroups based on the hilbert c module structure we study non stationary monotone quantum stochastic processes and general markov processes constructed from quantum dynamical hemigroups indexed by a totally ordered set we prove that the quantum stochastic monotone process implementing the weakly covariant process described by a covariant quantum dynamical hemigroup with respect to a symmetry semigroup is again covariant in the strong sense

discrete reproducing kernel hilbert spaces sampling and distribution of dirac masses we study reproducing kernels and associated reproducing kernel hilbert spaces rkhss h over infinite discrete and countable sets v in this setting we analyze in detail the distributions of the corresponding dirac point masses of v illustrations include certain models from neural networks an extreme learning machine elm is a neural network configuration in which a hidden layer of weights are randomly sampled and where the object is then to compute resulting output for rkhss h of functions defined on a prescribed countable infinite discrete set v we characterize those which contain the dirac masses x for all points x in v further examples and applications where this question plays an important role are i discrete brownian motion hilbert spaces i e discrete versions of the cameron martin hilbert space ii energy hilbert spaces corresponding to graph laplacians where the set v of vertices is then equipped with a resistance metric and finally iii the study of gaussian free fields

stationary stochastic processes in a group system in this paper we discuss positive definite functions of locally compact groups into the predual of von neumann algebras or c algebras for a predual valued or a c algebra valued weakly continuous covariant positive definite function the corresponding covariant representation in a covariant group system is constructed we consider some generalization of the time domain of stochastic processes to a locally compact group and we use covariant representations in a group system to obtain invariant stationary stochastic processes induced from predual valued covariant positive definite functions and covariant stationary stochastic processes induced from c algebra valued covariant positive definite functions