schur recursions error formulas and convergence of rational estimators for stationary stochastic sequences an exact and approximate realization theory for estimation and model filters of second order stationary stochastic sequences is presented the properties of j lossless matrices as a unifying framework are used necessary and sufficient conditions for the exact realization of an estimation filter and a model filter as a submatrix of a j lossless system are deduced an extension of the so called schur algorithm yields an approximate j lossless realization based on partial past information about the process the geometric properties of such partial realizations and their convergence are studied finally connections with the nevanlinna pick problem are made and how the techniques presented constitute a generalization of many aspects of the levinson szego theory of partial realizations is shown as a consequence generalized recursive formulas for reproducing kernels and christoffel darboux formulas are obtained in this paper the scalar case is considered the matrix case will be considered in a separate publication

brune sections in the non stationary case rational j inner valued functions which are j inner with respect to the unit circle j being a matrix which is both self adjoint and unitary play an important role in interpolation theory and are extensively utilized in signal processing for filtering purposes and in control for minimal sensitivity h feedback any such function is a product of three kinds of elementary factors each of them having a unique singularity outside the unit disk inside the unit disk and on the unit circle respectively counterparts of the first kind have already been studied in the context of non stationary systems when analytic functions are replaced by upper triangular operators the purpose of the present work is to study the non stationary analogues of the factors of the third kind one main difficulty is that one leaves the realm of bounded upper triangular operators and considers unbounded operators yet as is the case for a number of special clases of non stationary systems all the systems under consideration are finitely specified and the computations are done recursively on a finite set of state space data we consider the particular case where the operator given is of the ivi type that is it is time invariant both for small and large indices and is time varying in between the theory results in a rather general factorization theorem that generalizes the time invariant case to finitely specified time varying systems

the schur algorithm applied to the one dimensional continuous inverse scattering problem the one dimensional continuous inverse scattering problem can be solved by the schur algorithm in the discrete time domain using sampled scattering data the sampling rate of the scattering data should be increased to reduce the discretization error but the complexity of the schur algorithm is proportional to the square of the sampling rate to improve this tradeoff between the complexity and the accuracy we propose a schur algorithm with the richardson extrapolation sare the asymptotic expansion of the schur algorithm necessary for the richardson extrapolation is derived in powers of the discretization step which shows that the accuracy order with respect to the discretization step of the schur algorithm is 1 the accuracy order of the sare with the n step richardson extrapolation is increased to n 1 with comparable complexity to the schur algorithm therefore the discretization error of the schur algorithm can be decreased in a computationally efficient manner by the sare

on a generalization of the szego levinson recurrence and its application in lossless inverse scattering predictor polynomials corresponding to nested toeplitz matrices are known to be connected by the szego levinson recurrence relation a generalization of that result where the relevant reduction process for toeplitz matrices of decreasing order is defined by an elementary one parameter linear transformation is addressed the descending and ascending versions of the corresponding generalized szego levinson recurrence relations are discussed in detail in particular these relations are shown to be essentially the same as the extraction formulas for canonical schur and brune sections in the dewilde dym 1984 recursive solution of the lossless inverse scattering problem some extensions of the levinson algorithm for linear prediction and of the schur cohn algorithm for polynomial stability test are presented

local entropy averages and projections of fractal measures we show that for families of measures on euclidean space which satisfy an ergodic theoretic form of self similarity under the operation of re scaling the dimension of linear images of the measure behaves in a semi continuous way we apply this to prove the following conjecture of furstenberg let m n be integers which are not powers of the same integer and let x y be closed subsets of the unit interval which are invariant respectively under times m mod 1 and times n mod 1 then for any non zero t dim x ty min 1 dim x dim y a similar result holds for invariant measures and gives a simple proof of the rudolph johnson theorem our methods also apply to many other classes of conformal fractals and measures as another application we extend and unify results of peres shmerkin and nazarov and of moreira concerning projections of products self similar measures and gibbs measures on regular cantor sets we show that under natural irreducibility assumptions on the maps in the ifs the image measure has the maximal possible dimension under any linear projection other than the coordinate projections we also present applications to bernoulli convolutions and to the images of fractal measures under differentiable maps

numerical stability properties of a qr based fast least squares algorithm the numerical stability of a recent qr based fast least squares algorithm is established from a backward stability perspective a stability domain approach applicable to any least squares algorithm constructed from the set of reachable states in exact arithmetic is presented the error propagation question is shown to be subordinate to a backward consistency constraint which requires that the set of numerically reachable variables be contained within the stability domain associated to the algorithm this leads to a conceptually lucid approach to the numerical stability question which frees the analysis of stationary assumptions on the filtered sequences and obviates the tedious linearization methods of previous approaches moreover initialization phenomena and considerations related to poorly exciting inputs admit clear interpretations from this perspective the algorithm under study is proved in contrast to many fast algorithms to be minimal

a frequency domain condition for the physical realizability of linear quantum systems a recently emerging approach to the feedback control of linear quantum systems involves the use of a controller which itself is a quantum linear system this approach to quantum feedback control referred to as coherent quantum feedback control has the advantage that it does not destroy quantum information is fast and has the potential for efficient implementation an important issue which arises both in the synthesis of linear coherent quantum controllers and in the modeling of linear quantum systems is the issue of physical realizability this issue relates to the property of whether a given set of linear quantum stochastic differential equations corresponds to a physical quantum system satisfying the laws of quantum mechanics under suitable assumptions the paper shows that the question of physical realizability is equivalent to a frequency domain j j unitary condition this is important in controller synthesis since it is the transfer function matrix of the controller which determines the closed loop system behavior

on fuzzy measures defined by seminormed fuzzy integrals and semiconormed fuzzy integrals abstract in this paper we show that for the continuous t seminorms and continuous t semiconorms the properties of null addivity pseudo null addivity autocontinuity and pseudo autocontinuity of fuzzy measures g can be preserved by the fuzzy measures g h and g h defined by seminormed fuzzy integrals and semiconormed fuzzy integrals respectively at most under the conditions that x y 0 x y 0 and x y x y g can be preserved by the fuzzy measures g h under the condition that x y 0 x y 0

lossless inverse scattering digital filters and estimation theory methods to construct rational solutions of the lossless inverse scattering lis problem for one port passive digital systems are described the first method is recursive and can be viewed as a generalization of the celebrated schur algorithm the second method is global and leads to a parametrization of what we call fundamental solutions from which all lis solutions may be constructed quite a few classical problems in estimation theory and network theory may be viewed as special cases of the lis problem with each fundamental solution there is a solution of a corresponding estimation problem leading to a prediction and a modeling filter for a given stochastic process

generalized szego theory in frequency analysis abstract a sequence of positive borel measures with infinite support on the unit circle is given converging in the weak star topology to a measure with support consisting of a finite number n 0 of mass points the sequences of monic rational functions orthogonal with respect to inner products determined by the measures are studied with the focus being on the asymptotic behavior of the zeros the main result is twofold for a fixed n n 0 n 0 of the zeros interesting zeros tend to the mass points of the limiting measure while the remaining n n 0 zeros uninteresting zeros may not converge at all however by considering subsequences limits may be obtained and the remaining zeros and their limits will be located in a closed disk z r

selection of a single uniquely specifiable h controller in the chain scattering framework this paper studies the notion of central h controllers in the chain scattering framework since in the chain scattering framework the h control problem is equivalent to a factorisation problem that yields non unique factors it is important to somehow pin down these factors so that the resulting central controller is uniquely defined and corresponds to the central and minimum entropy controller frequently discussed in the literature in so doing we devise a procedure for the selection of a single uniquely specifiable h controller in the chain scattering framework the chain scattering framework is important because it has a distinct advantage over state space based methods for some applications in that it poses and solves the h control problem entirely in an operator theoretic setting thereby allowing us to contemplate direct changes of the frequency domain symbols rather than changes in the corresponding state space matrices

on orthogonal systems of rational functions on the unit circle and polynomials orthogonal with respect to varying measures abstract we study the recurrence relations of rational functions that are orthonormal on the unit circle and have all their poles in a given sequence of points outside the unit disk these relations enable us to find the recurrence relations and the ratio asymptotic behaviors for the orthonormal polynomials with respect to varying weights d n and d n 1 at the end of this paper we also consider the convergence theorem for multipoint pade approximations of markov type functions

favard theorem for reproducing kernels abstract consider for n 0 1 the nested spaces l n of rational functions of degree n at most with given poles 1 i i i 1 n let l 0 l n given a finite positive measure on the unit circle we associate with it an inner product on l by g g d suppose kn z w is the reproducing kernel for l n i e z k n z w w for all l n w then it is known that they satisfy a coupled recurrence relation in this paper we shall prove a favard type theorem which says that if you have a sequence of kernel functions kn z w which are generated by such a recurrence then there will be a measure supported on the unit circle so that kn is the reproducing kernel for ln the measure is unique under certain extra conditions on the points i

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

non euclidean geometrical aspects of the schur and levinson szego algorithms in this paper we address non euclidean geometrical aspects of the schur and levinson szego algorithms we first show that the lobachevski geometry is by construction one natural geometrical environment of these algorithms since they necessarily make use of automorphisms of the unit disk we next consider the algorithms in the particular context of their application to linear prediction then the schur resp levinson szego algorithm receives a direct resp polar spherical trigonometry st interpretation which is a new feature of the classical duality of both algorithms

diagonal pivoting for partially reconstructible cauchy like matrices with applications to toeplitz like linear equations and to boundary rational matrix interpolation problems abstract in an earlier paper we exploited the displacement structure of cauchy like matrices to derive for them a fast o n 2 implementation of gaussian elimination with partial pivoting one application is to the rapid and numerically accurate solution of linear systems with toeplitz like coefficient matrices based on the fact that the latter can be transformed into cauchy like matrices by using the fast fourier sine or cosine transform however symmetry is lost in the process and the algorithm given is not optimal for hermitian coefficient matrices in this paper we present a new fast o n 2 implementation of symmetric gaussian elimination with partial diagonal pivoting for hermitian cauchy like matrices and show how to transform hermitian toeplitz like matrices to hermitian cauchy like matrices obtaining algorithms that are twice as fast as those in the earlier work numerical experiments indicate that in order to obtain not only fast but also numerically accurate methods it is advantageous to explore the important case in which the corresponding displacement operators have nontrivial kernels this situation gives rise to what we call partially reconstructible matrices which are introduced and studied in the present paper we extend the transformation technique and the generalized schur algorithms i e fast displacement based implementations of gaussian elimination to partially reconstructible matrices we show by a variety of computed examples that the incorporation of diagonal pivoting methods leads to high accuracy we focused in this paper on the design of new numerically reliable algorithms for hermitian toeplitz like matrices however the proposed algorithms have other important applications in particular we briefly describe how they recursively solve a boundary interpolation problem for j unitary rational matrix functions

similarity between the mandelbrot set and julia sets the mandelbrot setm is self similar about any misiurewicz pointc in the sense that if we examine a neighborhood ofc inm with a very powerful microscope and then increase the magnification by a carefully chosen factor the picture will be unchanged except for a rotation the corresponding julia setj c is also self similar in the same sense with the same magnification factor moreover the two setsm andj c are similar in the sense that if we use a very powerful microscope to look atm andj c both focused atc the structures we see look like very much the same

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

on the null additivity and the autocontinuity of a fuzzy measure abstract in this paper further research on the null additivity and autocontinuity of fuzzy measures is made we prove that autocontinuity from below and autocontinuity from above of a fuzzy measure are equivalent when this fuzzy measure is finite furthermore if the space x is countable then the null additivity is equivalent to the autocontinuity

inner outer factorization and the inversion of locally finite systems of equations abstract we consider the problem of computing the inverse of a large class of infinite systems of linear equations which are described by a finite set of data the class consists of equations in which the linear operator is represented by a discrete time varying dynamical system whose local state space is of finite dimension at each time point k and which reduces to time invariant systems for time points k in this generalization of classical matrix inversion theory inner outer factorizations of operators play the role that qr factorization plays in classical linear algebra numerically they lead to so called square root implementations for which attractive algorithms can be derived which do not require the determination of spurious multiple eigenvalues as would be the case if the problem was converted to a discrete time riccati equation by squaring we give an overview of the theory and the derivation of the main algorithms the theory contains both the standard lti case and the case of a finite set of linear equations as special instances a particularly instance of which is called matrices of low hankel rank recently sometimes called quasi separable matrices however in the general case considered here new phenomena occur which are not observed in these classical cases namely the occurrence of defect spaces we describe these and give an algorithm to compute them as well in all cases the algorithms given are linear in the amount of data

multiple nevanlinna pick interpolation with both interior and boundary data and its connection with the power moment problem abstract the so called hankel vector approach is used to handle three boundary versions of the multiple nevanlinna pick interpolation problem in the nevanlinna class n involving both interior and boundary data it turns out that each of these boundary interpolation problems can be reduced to what amounts to a certain truncated standard or nonstandard hamburger moment problem associated with the hankel vector of the former with some possible constraint on distribution functions that assign no mass to any of the real nodes in particular this leads to solvability criteria for each of these interpolation problems and the description of solutions by using results from theory of moments

applications of the notion of analytic center in approximation estimation problems abstract a new method for solving a set of linear inequalities based on a generalization of the central maximum entropy solution of moment problems is applied to problems of numerical approximation cubature and filtering

past input reconstruction in fast least squares algorithms this paper solves the following problem given the computed variables in a fast least squares prediction algorithm determine all past input sequences that would have given rise to the variables in question this problem is motivated by the backward consistency approach to numerical stability in this algorithm class the set of reachable variables in exact arithmetic is known to furnish a stability domain our problem is equivalent to a first and second order interpolation problem introduced by mullis and roberts 1976 and studied by others our solution differs in two respects first relations to classical interpolation theory are brought out which allows us to parametrize all solutions by products of our formulation are correct necessary and sufficient conditions for the problem to be solvable in contrast to previous works whose claimed sufficient conditions are shown to fall short second our solution obtains any valid past input as the impulse response of an appropriately constrained orthogonal filter whose rotation parameters derive in a direct manner from the computed variables in a fast least squares prediction algorithm formulas showing explicitly the form of all valid past inputs should facilitate the study of what past input perturbation is necessary to account for accumulated arithmetic errors in this algorithm class this in turn is expected to have an impact in studying accuracy aspects in fast least squares algorithms

unitary interpolants factorization indices and infinite hankel block matrices abstract the existence uniqueness and construction of unitary n n matrix valued functions j j j in wiener like algebras on the circle with prescribed matrix fourier coefficients j j for j 0 are studied in particular if j then such an exists with j if and only if 0 1 where v denotes the infinite block hankel matrix j k v j k 0 1 acting in the sequence space l n 2 one of the main results is that the nonnegative factorization indices of every such are uniquely determined by the given data in terms of the dimensions of the kernels of i v v whereas the negative factorization indices are arbitrary it is also shown that there is a unique such if and only if the data forces all the factorization indices to be nonnegative and simple conditions for that and a formula for in terms of certain schmidt pairs of 0 are given the results depend upon a fine analysis of the structure of the kernels of i v v and of the one step extension problem of adamjan arov and krein funct anal appl 2 1968 1 18 isometric interpolants for the nonsquare case are also considered

strong and weak convergence of orthogonal systems of rational functions on the unit circle abstract pan k strong and weak convergence of orthogonal systems of rational functions on the unit circle journal of computational and applied mathematics 46 1993 427 436 we study weak and strong convergence of rational functions that are orthonormal on the unit circle and have all their poles in a given sequence of points outside the unit disk the erd s measures have been characterized as well

on fuzzy measures defined by fuzzy integrals on definit une mesure floue comme une fonction ensemble sur 7b f qui est une algebre de sous ensembles de l ensemble x

on the existence and construction of solutions to the partial lossless inverse scattering problem with applications to estimation theory the partial lossless inverse scattering plis problem which plays a fundamental role in the solution of a alpha stationary estimation problems is considered some open problems connected with this theory are treated necessary and sufficient conditions for a hermitian matrix to be positive say a covariance matrix are derived in terms of its alpha stationary wave representation a generalization of the recursive construction of plis solutions is proposed it is shown that each invariant subspace of the backward shift operator has a corresponding solution of the plis problem provided the operator that generalized the classical pick matrix is positive definite it is also shown that under fairly general conditions all solutions of the h sup infinity type are essentially obtained in this way and a characterization of the residual wave quantities generated by the scatterer is given stochastic models and forward and backward order recursive innovation filters are derived

realization of power spectra from partial covariance sequences in this paper we address the problem of realization of a spectral density function from incomplete information about the underlying stochastic process the standing assumption is the availability of an incomplete partial sequence of covariance samples of the process we study the set of rational extensions of this finite sequence to an infinite covariance function that agrees with the available samples the classical theory of orthogonal polynomials with respect to the unit circle and the theory of moments have been utilized extensively in a variety of engineering problems including the one we are dealing with these have been known to provide a unifying framework for a variety of current spectral estimation techniques maximum entropy method pisarenko s harmonic decomposition etc in this work we consider and study the set of all covariance realizations of dimension lower than or equal to the length of the partial sequence and equal to the dimension of the maximum entropy realization the me solution is a point in this set other points correspond to pole zero models a general formula is obtained for recursively updated pole zero models of dimension increasing with the data record information about the zeros is obtained from the asymptotic behavior of the partial autocorrelation coefficients our approach combines techniques from the theory of orthogonal polynomials on the unit circle the theory of moments and also techniques from degree theory topology our objective is to develop a theory that will provide a frame for constructing recursively pole zero realizations of increasing dimension

a boundary nevanlinna pick problem for a class of analytic matrix valued functions in the unit ball abstract we solve a tangential boundary interpolation problem with a finite number of interpolating points for a multivariable analogue of the schur class the description of all solutions is parametrized in terms of a linear fractional transformation whose entries are given explicitly in terms of the interpolation data

the scenery flow for hyperbolic julia sets we define the scenery flow space at a point z in the julia set j of a hyperbolic rational map t c c with degree at least 2 and more generally for t a conformal mixing repellor we prove that for hyperbolic rational maps except for a few exceptional cases listed below the scenery flow is ergodic we also prove ergodicity for almost all conformal mixing repellors here the statement is that the scenery flow is ergodic for the repellors which are not linear nor contained in a finite union of real analytic curves and furthermore that for the collection of such maps based on a fixed open set u the ergodic cases form a dense open subset of that collection scenery flow ergodicity implies that one generates the same scenery flow by zooming down towards almost every z with respect to the hausdorff measure hd where d is the dimension of j and that the flow has a unique measure of maximal entropy for all conformal mixing repellors the flow is loosely bernoulli and has topological entropy at most d moreover the flow at almost every point is the same up to a rotation and so as a corollary one has an analogue of the lebesgue density theorem for the fractal set giving a different proof of a theorem of falconer

a tangential interpolation problem on the distinguished boundary of the polydisk for the schur agler class abstract we study a tangential interpolation problem with an arbitrary set of interpolating points on the distinguished boundary of the unit polydisk for schur agler class the description of all solutions is parametrized in terms of a linear fractional transformation

rational approximation in linear systems and control abstract in this paper we want to describe some examples of the active interaction that takes place at the border of rational approximation theory and linear system theory these examples are mainly taken from the period 1950 1999 and are described only at a skindeep level in the simplest possible scalar case we give comments on generalizations of these problems and how they opened up new ranges of research that after a while lived their own lives we also describe some open problems and future work that will probably continue for some years after 2000

asymptotic properties of zeros of orthogonal rational functions abstract a sequence x m m 0 1 2 of real numbers gives rise to an absolutely continuous measure n t on the unit circle given by d t n d 12 n 1 m 0 x m t m e im 2 where n is a natural number t 0 1 each measure n t determines a sequence n n t of monic orthogonal rational functions with prescribed poles outside the closed unit disk in particular these rational functions may be polynomials a measure 0 with support consisting of the points 1 n0 on the unit circle is given let n n0 under a suitable condition on weak convergence of the measures and with a proper ordering of the zeros zk n n t of n n t it can be shown that lim t 1 lim t z k n n t k k 1 n 0 and that there exists a positive number n z k n n t n for k n0 1 n for all n and all t 0 1 the theory is applied to the problem of determining unknown frequencies in a trigonometric signal x m ij iajeimwj m 0 1 2