cluster values of holomorphic functions of bounded type we study the cluster value theorem for hb x the frechet algebra of holomorphic functions bounded on bounded sets of x we also describe the size of fibers of the spectrum of hb x our results are rather complete whenever x has an unconditional shrinking basis and for x 1 as a byproduct we obtain results on the spectrum of the algebra of all uniformly continuous holomorphic functions on the ball of 1

interpolation in the nevanlinna and smirnov classes and harmonic majorants abstract we consider a free interpolation problem in nevanlinna and smirnov classes and find a characterization of the corresponding interpolating sequences in terms of the existence of harmonic majorants of certain functions we also consider the related problem of characterizing positive functions in the disk having a harmonic majorant an answer is given in terms of a dual relation which involves positive measures in the disk with bounded poisson balayage we deduce necessary and sufficient geometric conditions both expressed in terms of certain maximal functions

mse bounds with affine bias dominating the cramer rao bound in continuation to an earlier work we further develop bounds on the mean squared error mse when estimating a deterministic parameter vector thetas0 in a given estimation problem as well as estimators that achieve the optimal performance the traditional cramer rao cr type bounds provide benchmarks on the variance of any estimator of thetas0 under suitable regularity conditions while requiring a priori specification of a desired bias gradient to circumvent the need to choose the bias which is impractical in many applications it was suggested in our earlier work to directly treat the mse which is the sum of the variance and the squared norm of the bias while previously we developed mse bounds assuming a linear bias vector here we study in the same spirit affine bias vectors we demonstrate through several examples that allowing for an affine transformation can often improve the performance significantly over a linear approach using convex optimization tools we show that in many cases we can choose an affine bias that results in an mse bound that is smaller than the unbiased cr bound for all values of thetas0 furthermore we explicitly construct estimators that achieve these bounds in cases where an efficient estimator exists by performing an affine transformation of the standard maximum likelihood ml estimator this leads to estimators that result in a smaller mse than ml for all possible values of thetas0

stability of asymptotics of christoffel darboux kernels we study the stability of convergence of the christoffel darboux kernel associated with a compactly supported measure to the sine kernel under perturbations of the jacobi coefficients of the measure we prove stability under variations of the boundary conditions and stability in a weak sense under l 1 and random l 2 diagonal perturbations we also show that convergence to the sine kernel at x implies that x 0

preservation of a c spectrum for random decaying perturbations of square summable high order variation we consider random self adjoint jacobi matrices of the form j u n an u n 1 bn u n an 1 u n 1 on l2 n where an 0 and bn r are sequences of random variables on a probability space dp such that there exists q n such that for any l n 2l al al q and 2l 1 bl bl q are independent random variables of zero mean satisfying n 1 n2 dp let jp be the deterministic periodic of period q jacobi matrix whose coefficients are the mean values of the corresponding entries in j we prove that for a e the a c spectrum of the operator j equals to and fills the spectrum of jp if moreover n 1 n4 dp then for a e the spectrum of j is purely absolutely continuous on the interior of the bands that make up the spectrum of jp

multi dimensional schrodinger operators with some negative spectrum abstract we obtain conditions on the negative spectra of schrodinger operators with potentials v and v which guarantee that the positive real line is covered by the absolutely continuous spectrum

a comparison of snr estimation techniques for the awgn channel the performances of several signal to noise ratio snr estimation techniques reported in the literature are compared to identify the best estimator the snr estimators are investigated by the computer simulation of baseband binary phase shift keying psk signals in real additive white gaussian noise awgn and baseband 8 psk signals in complex awgn the mean square error is used as a measure of performance in addition to comparing the relative performances the absolute levels of performance are also established the simulated performances are compared to a published cramer rao bound crb for real awgn and a crb for complex awgn that is derived here some known estimator structures are modified to perform better on the channel of interest estimator structures for both real and complex channels are examined

stability of spectral types for jacobi matrices under decaying random perturbations abstract we study stability of spectral types for semi infinite self adjoint tridiagonal matrices under random decaying perturbations we show that absolutely continuous spectrum associated with bounded eigenfunctions is stable under hilbert schmidt random perturbations we also obtain some results for singular spectral types

absolutely continuous spectrum of dirac operators with square integrable potentials we show that the absolutely continuous part of the spectral function of the one dimensional dirac operator on a half line with a constant mass term and a real square integrable potential is strictly increasing throughout the essential spectrum 1 1 the proof is based on estimates for the transmission coefficient for the full line scattering problem with a truncated potential and a subsequent limiting procedure for the spectral function furthermore we show that the absolutely continuous spectrum persists when an angular momentum term is added thus also establishing the result for spherically symmetric dirac operators in higher dimensions

nonlinear fourier transform towards the construction of nonlinear fourier modes we study a version of the nonlinear fourier transform associated with zs akns systems this version is suitable for the construction of nonlinear analogues of fourier modes and for the perturbation theoretic study of their superposition we provide an iterative scheme for computing the inverse of our transform the relevant formulae are expressed in terms of bell polynomials and functions related to them in order to prove the validity of our iterative scheme we show that our transform has the necessary analytic properties we show that up to order three of the perturbation parameter the nonlinear fourier mode is a complex sinusoid modulated by the second bernoulli polynomial we describe an application of the nonlinear superposition of two modes to a problem of transmission through a nonlinear medium

cramer rao bounds for snr estimation of oversampled linearly modulated signals most signal to noise ratio snr estimators use the receiver matched filter output sampled at the symbol rate an approach which does not preserve all information in the analog waveform due to aliasing thus it is relevant to ask whether avoiding aliasing could improve snr estimation to this end we compute the corresponding data aided da and nondata aided nda cramer rao bounds crbs we adopt a novel dual filter framework which is shown to be information preserving under suitable conditions and considerably simplifies the analysis it is shown that the crb can be substantially reduced by exploiting any available excess bandwidth depending on the modulation scheme the snr range and the estimator type da or nda

converse growth estimates for odes with slowly growing solutions let f 1 f 2 be linearly independent solutions of f af 0 where the coefficient a is an analytic function in the open unit disc mathbb d of the complex plane mathbb c it is shown that many properties of this differential equation can be described in terms of the subharmonic auxiliary function u log f 1 f 2 for example the case when sup z in mathbb d a z 1 z 2 2 infty and f 1 f 2 is normal is characterized by the condition sup z in mathbb d nabla u z 1 z 2 infty different types of blaschke oscillatory equations are also described in terms of harmonic majorants of u even if f 1 f 2 are bounded linearly independent solutions of f af 0 it is possible that sup z in mathbb d a z 1 z 2 2 infty or f 1 f 2 is non normal these results relate to sharpness discussion of recent results in the literature and are succeeded by a detailed analysis of differential equations with bounded solutions analogues for the nevanlinna class are also considered by taking advantage of nevanlinna interpolating sequences it is shown that instead of considering solutions with prescribed zeros it is possible to construct a bounded solution of f af 0 in such a way that it solves an interpolation problem natural to bounded analytic functions while a z 2 1 z 2 3 dm z remains to be a carleson measure

best constants for the riesz projection we prove the following inequality with a sharp constant p f l p t csc p f lp t f lp t where 1 p and p lp t hp t is the riesz projection onto the hardy space hp t on the unit circle t in other words the angle between the analytic and co analytic subspaces of lp t equals p where p max p pp 1 this was conjectured in 1968 by i gohberg and n krupnik we also prove an analogous inequality in the nonperiodic case where p f f 1 r ff is the half line fourier multiplier on r similar weighted inequalities with sharp constants for lp r x 1 p 1 are obtained in the multidimensional case our results give the norm of the half space fourier multiplier on rn

the absolutely continuous spectrum of one dimensional schrodinger operators with decaying potentials we investigate one dimensional schrodinger operators with asymptotically small potentials it will follow from our results that if with then is an essential support of the absolutely continuous part of the spectral measure we also prove that if then the spectrum is purely absolutely continuous on these results are optimal

solutions spectrum and dynamics for schrodinger operators on infinite domains 1 introduction and main results in this paper we investigate the relations between the rate of decay of solutions of schrodinger equations continuity properties of spectral measures of the corresponding operators and dynamical properties of the corresponding quantum systems the first main result of this paper shows that in great generality certain upper bounds on the rate of growth of l 2 norms of generalized eigenfunctions over expanding balls imply certain minimal singularity of the spectral measures consider an operator h v defined by the differential expression h v v x

paths of inner related functions we characterize the connected components of the subset cn of h formed by the products bh w here b is carleson newman blaschke product and h h is an invertible function we use this result to show that except for finite blaschke products no inner function in the little bloch space is in the closure of one of these components our main result says that every inner function can be connected with an element of cn within the set of products uh w hereu is inner and h is invertible we also study some of these issues in the context of douglas algebras

sum rules and spectral measures of schrodinger operators with l 2 potentials necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a half line schrodinger operator with square integrable potential

analytic projections corona problem and geometry of holomorphic vector bundles 9 d d and d operators d z d i a a normalized laplacian a dd norm since we are dealing with matrixand op er at orvalued functions we will use the symbol usually with a subscript for the norm in a functions space while is used for the norm in the underlying vector operator space thus for a vectorvalued function the symbol 2 denotes its l2 norm but the symbol stands for

speech enhancement using a minimum mean square error short time spectral amplitude estimator this paper focuses on the class of speech enhancement systems which capitalize on the major importance of the short time spectral amplitude stsa of the speech signal in its perception a system which utilizes a minimum mean square error mmse stsa estimator is proposed and then compared with other widely used systems which are based on wiener filtering and the spectral subtraction algorithm in this paper we derive the mmse stsa estimator based on modeling speech and noise spectral components as statistically independent gaussian random variables we analyze the performance of the proposed stsa estimator and compare it with a stsa estimator derived from the wiener estimator we also examine the mmse stsa estimator under uncertainty of signal presence in the noisy observations in constructing the enhanced signal the mmse stsa estimator is combined with the complex exponential of the noisy phase it is shown here that the latter is the mmse estimator of the complex exponential of the original phase which does not affect the stsa estimation the proposed approach results in a significant reduction of the noise and provides enhanced speech with colorless residual noise the complexity of the proposed algorithm is approximately that of other systems in the discussed class

cluster values of analytic functions on a banach space we investigate uniform algebras of bounded analytic functions on the unit ball of a complex banach space we prove several cluster value theorems relating cluster sets of a function to its range on the fibers of the spectrum of the algebra these lead to weak versions of the corona theorem for l2 and for c0 in the case of the open unit ball of c0 we solve the corona problem whenever all but one of the functions comprising the corona data are uniformly approximable by polynomials in functions in c 0

finite gap potentials and wkb asymptotics for one dimensional schrodinger operators consider the schrodinger operator h d2 dx2 v x with power decaying potential v x o x we prove that a previously obtained dimensional bound on exceptional sets of the wkb method is sharp in its whole range of validity the construction relies on pointwise bounds on finite gap potentials these bounds are obtained by an analysis of the jacobi inversion problem on hyperelliptic riemann surfaces

generalized carleson newman inner functions we study a class of inner functions introduced by gorkin mortini and nikolski and motivated by banach algebra and functional calculus applications answering their question we produce a singular function that cannot be multiplied into this generalized carleson newman class of inner functions the class of functions satisfying the so called weak embedding property furthermore we give elementary proofs for some results on these classes that were obtained earlier using arguments related to the gelfand representation of h infty

approximation by ratios of bounded analytic functions let d be the unit disk andebe a compact subset of theh maximal ideal space m h we show that any continuous functionfon an open neighborhoodu m h ofesuch thatf u d h u d can be uniformly approximated oneby ratiosh g whereh g h andgis zero free one we also characterize those setsefor whichh gcan be replaced byhfor everyf finally we prove that any inner functionucan be written as a rational function of interpolating blaschke products it follows thatucan be uniformly approximated by ratios of interpolating blaschke products on any compact sete m h whereuis zero free

the matrix valued hp corona problem in the disk and polydisk abstract in this paper we consider the matrix valued h p corona problem in the disk and polydisk the result for the disk is rather well known and is usually obtained from the classical carleson corona theorem by linear algebra our proof provides a streamlined way of obtaining this result and allows one to get a better estimate on the norm of the solution in particular we were able to improve the estimate found in the recent work of trent in j funct anal 189 2002 267 282 note that the solution of the h matrix corona problem in the disk can be easily obtained from the h 2 corona problem either by factorization or by the commutant lifting theorem the h p corona problem in the polydisk was originally solved by lin in bull sci math 110 2 1986 69 84 trans amer math soc 341 1994 371 375 the solution used koszul complexes and was rather complicated because one had to consider higher order equations our proof is more transparent and it improves upon lin s result in several ways first we deal with the more general matrix corona problem second we were able to show that the norm of the solution is independent of the number of generators additionally we illustrate that the norm of the solution of the h 2 corona problem in the polydisk d n grows at most proportionally to n our approach is based on one that was originated by andersson in math z 201 1989 121 130 in the disk it essentially depends on green s theorem and duality to obtain the estimate in the polydisk we use riesz projections to reduce the problem to the disk case

uniformly improving maximum likelihood snr estimation of known signals in gaussian channels the signal to noise ratio snr estimation problem is considered for an amplitude modulated known signal in gaussian noise the benchmark method is the maximum likelihood estimator mle whose merits are well documented in the literature in this work an affinely modified version of the mle ammle that uniformly outperforms over all snr values the traditional mle in terms of the mean square error mse is obtained in closed form however construction of an ammle whose mse is lower at every snr than the unbiased cramer rao bound ucrb is shown to be infeasible in light of this result the ammle construction rule is modified to provision for an a priori known set s where the snr lies and the mse enhancement target is pursued within s the latter is realized through proper extension of an existing framework due to eldar which settles the design problem by solving a semidefinite program the analysis is further extended to the general case of vector signal models numerical results show that the proposed design demonstrates enhancement of the mse for all the considered cases

the spectrum of differential operators of order 2n with almost constant coefficients abstract we discuss the spectral properties of higher order ordinary differential operators if the coefficients differ from constants by small perturbations then the spectral properties are preserved in this context small perturbations are either short range i e integrable or long range but slowly varying this generalizes classical results on second order operators our approach relies on an analysis of the associated differential equations with the help of uniform asymptotic integration techniques

toeplitz corona and the douglas property for free functions abstract the well known douglas lemma says that for operators a b on hilbert space that a a b b 0 implies b a c for some contraction operator c the result carries over directly to classical operator valued toeplitz operators simply by replacement of operator by toeplitz operator throughout free functions generalize the notion of free polynomials and formal power series and trace back to the work of j taylor in the 1970s they are of current interest in part because of their connections with free probability and engineering systems theory in this article for given free functions a and b with noncommutative domain k defined by free polynomial inequalities we obtain a sufficient condition in terms of the difference a a b b for the existence of a free function c taking contractive values on k such that b a c the connection to recent work of agler and mccarthy and their free toeplitz corona theorem is expounded

on zero curvature condition and fourier analysis we study a system of conserved quantities of the periodic klein gordon equation we obtain these quantities by means of a perturbation construction from a lax pair of the periodic sine gordon equation we show that for a suitable choice of values of the spectral parameter our conserved quantities have simple expressions in terms of the fourier coefficients of the initial data moreover they turn out to be the action variables this provides an interesting illustration of the role of the spectral parameter our perturbation construction also provides a new lax pair for the klein gordon equation and our action variables arise from this lax pair this turns out to be a special case k 2 of a more general lax pair for a certain k jet system of the sine gordon equation the structure algebra of this lax pair is the algebra of upper triangular k k block toeplitz matrices whose blocks are elements of the ad invariant functions on the lie group corresponding to the lie algebra are needed for the construction of the integrals these functions are not given by the spectra of the matrices they have to be constructed by other means

first kdv integrals and absolutely continuous spectrum for 1 d schrodinger operator we consider 1 d schrodinger operators on l 2 r with slowly decaying potentials under some conditions on the potential related to the first integrals of the kdv equation we prove that the a c spectrum of the operator coincides with the positive semiaxis and the singular spectrum is unstable examples show that for special classes of sparse potentials these results can not be improved

on the nonlinear fourier transform associated with periodic akns zs systems and its inverse we study the nonlinear fourier transform associated with the integrable nonlinear partial differential equations of akns zs type we show that is a real analytic operator between the appropriate hilbert spaces and that it has a real analytic local inverse near the origin we construct a convergent iterative scheme by means of which one can calculate the inverse to any desired degree of accuracy

maximal functions associated to filtrations abstract let t be a bounded linear or sublinear operator from lp y to lq x a maximal operator t f x supj t f yj x is associated to any sequence of subsets yj of y under the hypotheses that q p and the sets yj are nested we prove that t is also bounded classical theorems of menshov and zygmund are obtained as corollaries multilinear generalizations of this theorem are also established these results are motivated by applications to the spectral analysis of schrodinger operators

sum rules for jacobi matrices and their applications to spectral theory we discuss the proof of and systematic application of case s sum rules for jacobi matrices of special interest is a linear combination of two of his sum rules which has strictly positive terms among our results are a complete classification of the spectral measures of all jacobi matrices j for which j j 0 is hilbert schmidt and a proof of nevai s conjecture that the szego condition holds if j j 0 is trace class

wkb and spectral analysis of one dimensional schrodinger operators with slowly varying potentials consider a schrodinger operator on l 2 of the line or of a half line with appropriate boundary conditions if the potential tends to zero and is a finite sum of terms each of which has a derivative of some order in l 1 l p for some exponent p 2 then an essential support of the the absolutely continuous spectrum equals almost every generalized eigenfunction is bounded and satisfies certain wkb type asymptotics at infinity if moreover these derivatives belong to l p with respect to a weight x with 0 then the hausdorff dimension of the singular component of the spectral measure is strictly less than one

norm controlled inversions and a corona theorem for h quotient algebras let be an inner function on the unit disc d we give a description of those for which the quotient algebra h h has no corona with respect to the visible part of its spectrum that is for which m h h z d z 0 m h it happens that this property is equivalent to the norm controlled inversion property for h h as well as to a kind of weakened carleson type embedding theorem the quotient algebra a d h is also considered an interpretation of our main results in terms of model operators is given too

weak embedding property inner functions and entropy following gorkin mortini and nikolski we say that an inner function i in h infty mathbb d has the wep property if its modulus at a point z is bounded from below by a function of the distance from z to the zero set of i this is equivalent to a number of properties and we establish some consequences of this for h infty ih infty the bulk of the paper is devoted to wepable functions i e those inner functions which can be made wep after multiplication by a suitable blaschke product we prove that a closed subset e of the unit circle is of finite entropy i e is a beurling carleson set if and only if any singular measure supported on e gives rise to a wepable singular inner function as a corollary we see that singular measures which spread their mass too evenly cannot give rise to wepable singular inner functions furthermore we prove that the stronger property of porosity of e is equivalent to a stronger form of wepability easy wepability for the singular inner functions with support in e finally we find out the critical decay rate of masses of atomic measures with no restrictions on support guaranteeing that the corresponding singular inner functions are easily wepable

imbedded singular continuous spectrum for schrodinger operators 1 1 hv v x and some self adjoint boundary condition at zero the operator 1 1 describes a charged particle such as an electron in the electric field v x when v x is decaying quickly one expects the spectral and dynamical properties of hy to remain close to those of the free operator ho recall that with the operator 1 1 one can associate in a canonical way a spectral measure p see for example 6 2 which contains much information about the quantum system the classical result in this direction going back to the beginning of the century is that if v g l1 the spectral measure on the positive semi axis remains purely absolutely continuous in one dimension an absolutely continuous spectrum corresponds to the ballistic rate of propagation of the particle a natural question to ask is what are the critical rates of decay at which some changes in the spectral and dynamical properties of hy may happen as far as v x has any decay at all the essential spectrum coincides with 0 oo but the quality of the spectrum and dynamics may change in 1928 wigner and von neumann 27 showed that there exist potentials v x satisfying v x jsuch that hy has positive imbedded eigenvalue e 1 this is a purely quantum resonance phenomenon since v can be chosen arbitrarily small and the bound state originates from long range correlations in the potential rather than the usual confining effect naboko 17 and simon 25 provided constructions which show that much more drastic changes are possible if the potential decays arbitrarily slower than coulomb namely for any positive function h x which grows at infinity there exist potentials v such that v x yjgand hy has a dense set of positive eigenvalues due to some recent results 3 21 it is known that the absolutely continuous spectrum on the positive semi axis is preserved for v o c l x l 2 e and in fact for v g l2 8 more precisely for such potentials the absolutely continuous part of the spectral measure iac gives positive weight to any subset of 0 oo of positive lebesgue measure therefore the naboko

stable rank and approximation theorems in it is conjectured that for h the bass stable rank bsr is 1 and the topological stable rank tsr is 2 bsr h 1 if and only if for every f f2 e hx x h00 which is a corona pair i e there exist g g2 e h00 such that f gl f2g2 1 there exists a g e hco such that f f2g e h 1 the invertibles in h however it suffices to consider corona pairs f 1 f2 where f is a blaschke product it is also shown that there exists a g e h00 such that f f2g e exp h00 if and only if logf1 can be boundedly analytically defined on z e d if2 z i 0 tsr h 2 if and only if the corona pairs are uniformly dense in h00 x h however it suffices to show that the corona pairs are uniformly dense in pairs of blaschke products this condition would be satisfied if the interpolating blaschke products were uniformly dense in the blaschke products for b an inner function k h2 e bh2 is an h module via the compressed toeplitz operators cf pktf ik for f e h00 where tf is the 2 toeplitz operator tfg fg for g e h some stable rank questions can be recast as lifting questions for fi n c h 0 there exist gi hi n c h such that en1 f bgi hi 1 if and only if the compressed toeplitz operators cs 1 may be lifted to toeplitz operators tf 1 which generate b h2 as an ideal

one dimensional schrodinger operators with random decaying potentials we investigate the spectrum of the following random schrodinger operators h omega frac d 2 dt 2 a t f x t omega wheref xt is a markovian potential studied by the russian school 8 we completely describe the transition of the spectrum from pure point type to absolutely continuous type as the decreasing order ofa t grows this is an extension to a continuous case of the result due to delyon simon souillard 6 who deal with the lattice case

asymptotic green functions a generalized semiclassical approach for scattering by multiple barrier potentials we show how quantum mechanical barrier reflection and transmission coefficients and can be obtained from asymptotic green functions we exemplify our results by calculating such coefficients for the rosen morse rm potential for multiple barrier potentials v x jv j x where each v j goes to zero for x we derive the asymptotic green functions by a generalized semiclassical approximation which is based on the usual sum over classical paths considered only in the classically allowed regions and includes local quantum effects through the individual j and j the approach is applied to double rm potentials and to woods saxon barriers we obtain analytical expressions for the transmission and reflection probabilities of these potentials which are very accurate when compared with exact numerical calculations being much better than the usual wkb approximation finally we briefly discuss how to extend the present method to other kinds of potential

a new estimate for the vector valued corona problem abstract we improve an estimate of fuhrmann and vasyunin in the vector valued corona theorem

snr estimation in time varying fading channels signal to noise ratio snr estimation is considered for phase shift keying communication systems in time varying fading channels both data aided da estimation and nondata aided nda estimation are addressed the time varying fading channel is modeled as a polynomial in time inherent estimation accuracy limitations are examined via the cramer rao lower bound where it is shown that the effect of the channel s time variation on snr estimation is negligible a novel maximum likelihood ml snr estimator is derived for the time varying channel model in da scenarios where the estimator has a simple closed form solution the exact performance is evaluated both with correct and incorrect i e mismatched polynomial order in nda estimation the unknown data symbols are modeled as random and the marginal likelihood is used the expectation maximization algorithm is proposed to iteratively maximize this likelihood function simulation results show that the resulting estimator offers statistical efficiency over a wider range of scenarios than previously published methods

absolutely continuous spectrum of schrodinger operators with potentials slowly decaying inside a cone abstract for a large class of multi dimensional schrodinger operators it is shown that the absolutely continuous spectrum is essentially supported by 0 we require slow decay and mildly oscillatory behavior of the potential in a cone and can allow for arbitrary non negative bounded potential outside the cone in particular we do not require the existence of wave operators the result and method of proof extends previous work by laptev naboko and safronov

bounded solutions and absolute continuity of sturm liouville operators this work is a contribution to the theory of the absolutely continuous spectrum of one dimensional schriidinger operators i e sturm liouville operators generated by differential expressions of the form ru u qu in proofs of absolute continuity for such operators and also for the closely related dirac systems always asymptotic properties of solutions of zu lu for real or complex values of 1 have played an important role see for example 18 19 8 4 9 lo in these works and a number of others particular classes of potentials q were treated using properties of solutions in the course of the proof of absolute continuity somewhat more recently the connection between asymptotic properties of solutions and absolute continuity was given a theory i e results of the following form were proved let the solutions of zu lu satisfy some specific asymptotic property for a varying in some subset of r no or almost no a priori assumptions on q then operators generated by z are absolutely continuous in this set a first result of this type was given by carmona 3 who proved that local uniform boundedness of solutions together with their derivatives in some open a set implies pure absolute continuity in this set here we will mostly rely on the concept of non or racy of solutions which was shown by gilbert and pearson 7 12 61 to be a powerful tool in the investigation of the continuous spectrum of sturm liouville operators in section 2 we give some consequences of their results whi ch are useful in proofs of absolute continuity these consequences range from non existence of imbedded eigenvalues and the existence of absolutely continuous parts to pure absolute continuity and the existence of transient 210 0022 247x 92 5 00

a characterization of the schur property through the disk algebra abstract in this paper we give a new characterization of when a banach space e has the schur property in terms of the disk algebra we prove that e has the schur property if and only if a d e a d e w

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

trivial gleason parts and the topological stable rank of h we show that the subset of the maximal ideal space of h formed by the trivial gleason parts is totally disconnected a pair g h h is called a corona pair if inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i 0 where inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i is the open unit disc we prove that the set u 2 h of corona pairs is dense in h h

some geometric properties of disk algebras abstract in this paper we study some geometrical properties of certain classes of uniform algebras in particular the ball algebra a u b x of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex banach space x we prove that a u b x has k numerical index 1 for every k the lushness and also the ahsp moreover the disk algebra a d and more in general any uniform algebra whose choquet boundary has no isolated points is proved to have the polynomial daugavet property most of those properties are extended to the vector valued version a x of a uniform algebra a

inner functions bloch spaces and symmetric measures schwarz s lemma asserts that analytic mappings from the unit disc into itself decrease hyperbolic distances in this paper inner functions which decrease hyperbolic distances as much as possible when one approaches the unit circle are constructed actually it is shown that a quadratic condition governs the best decay of the hyperbolic derivative of an inner function this is related to a result of l carleson on the existence of singular symmetric measures as a consequence some results on composition operators are obtained bringing out the importance of the bloch spaces in this connection another consequence is a uniform way of producing singular measures which are simultaneously symmetric and kahane 1991 mathematics subject classification primary 30d50 secondary 30d45 26a30 47b38

wkb asymptotic behavior of almost all generalized eigenfunctions for one dimensional schrodinger operators with slowly decaying potentials we prove the wkb asymptotic behavior of solutions of the differential equation d2u dx2 v x u eu for a e e a where v v1 v2 v1 lp r and v2 is bounded from above with a lim supx v x while v 2 x lp r 1 p 2 these results imply that schrodinger operators with such potentials have absolutely continuous spectrum on a we also establish wkb asymptotic behavior of solutions for some energy dependent potentials

analysis and comparison of biased affine estimators affine biased estimation is particularly useful when there is some a priori knowledge on the parameters that can be exploited in adverse situations when the number of samples is low or the noise is high three different affine estimation strategies are discussed namely the deepest minimum criterion dmc the min max mm and the linear matrix inequality lmi strategies and closed form expressions are obtained for all of them for the case when the a priori knowledge is given in the form of ellipsoidal constraints on the parameter space and when the covariance matrix of the unbiased estimator is constant a relationship between affine estimation and bayesian estimation of the mean of a multivariate gaussian distribution with gaussian prior is established and it is shown how affine estimation theory can help in the choice of the gaussian prior distribution

the corona property in nevanlinna quotient algebras and interpolating sequences abstract let i be an inner function in the unit disk d and let n denote the nevanlinna class we prove that under natural assumptions bezout equations in the quotient algebra n i n can be solved if and only if the zeros of i form a finite union of nevanlinna interpolating sequences this is in contrast with the situation in the algebra of bounded analytic functions where being a finite union of interpolating sequences is a sufficient but not necessary condition an analogous result in the smirnov class is proved as well as several equivalent descriptions of blaschke products whose zeros form a finite union of interpolating sequences in the nevanlinna class

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

on the absolutely continuous spectrum of stark operators the stability of the absolutely continuous spectrum of the one dimensional stark operator under perturbations of the potential is discussed the focus is on proving this stability under minimal assumptions on smoothness of the perturbation a general criterion is presented together with some applications these include the case of periodic perturbations where we show that any perturbation v l 1 h 1 2 preserves the a c spectrum

absolutely continuous spectrum of one dimensional schrodinger operators and jacobi matrices with slowly decreasing potentials we prove that for any one dimensional schrodinger operator with potentialv x satisfying decay condition v x cx 3 4 e the absolutely continuous spectrum fills the whole positive semi axis the description of the set in on which the singular part of the spectral measure might be supported is also given analogous results hold for jacobi matrices

transient and recurrent spectrum abstract we deal primarily with spectral analysis of an abstract self adjoint operator h on a hilbert space h we propose a further refinement of the absolutely continuous subspace h ac into the transient absolutely continuous subspace h tac which is the closure of those with e ith o t n for all n and the recurrent absolutely continuous subspace h rac h ac h tac we discuss general features of this breakup in a subsequent paper we construct analytic almost periodic functions v on so that h d 2 dx 2 v x on l 2 has only recurrent absolutely continuous spectrum in the sense that h rac h

jacobi matrices with power like weights grouping in blocks approach the paper deals with jacobi matrices with weights k given by k k 1 k where 12 1 and limk k 0 the main question studied here concerns when the spectrum of the operator j defined by the jacobi matrix has absolutely continuous component covering the real line a sufficient condition is given for a positive answer to the above question the method used in the paper is based on a detailed analysis of generalized eigenvectors of j in turn this analysis relies on the so called grouping in blocks approach to a large product of the transfer matrices associated to j

uniformly improving the cramer rao bound and maximum likelihood estimation an important aspect of estimation theory is characterizing the best achievable performance in a given estimation problem as well as determining estimators that achieve the optimal performance the traditional cramer rao type bounds provide benchmarks on the variance of any estimator of a deterministic parameter vector under suitable regularity conditions while requiring a priori specification of a desired bias gradient in applications it is often not clear how to choose the required bias a direct measure of the estimation error that takes both the variance and the bias into account is the mean squared error mse which is the sum of the variance and the squared norm of the bias here we develop bounds on the mse in estimating a deterministic parameter vector x0 over all bias vectors that are linear in x0 which includes the traditional unbiased estimation as a special case in some settings it is possible to minimize the mse over all linear bias vectors more generally direct minimization is not possible since the optimal solution depends on the unknown x0 nonetheless we show that in many cases we can find bias vectors that result in an mse bound that is smaller than the cramer rao lower bound crlb for all values of x0 furthermore we explicitly construct estimators that achieve these bounds in cases where an efficient estimator exists by performing a simple linear transformation on the standard maximum likelihood ml estimator this leads to estimators that result in a smaller mse than the ml approach for all possible values of x0

on the absolutely continuous spectrum of one dimensional schrodinger operators with square summable potentials for continuous and discrete one dimensional schrodinger operators with square summable potentials the absolutely continuous part of the spectrum is essentially supported by 0 and 2 2 respectively this fact is proved by considering a priori estimates for the transmission coefficient